diff --git a/compiler/src/dotty/tools/dotc/ast/Desugar.scala b/compiler/src/dotty/tools/dotc/ast/Desugar.scala
index e5021e5b955e..0ae11faf502b 100644
--- a/compiler/src/dotty/tools/dotc/ast/Desugar.scala
+++ b/compiler/src/dotty/tools/dotc/ast/Desugar.scala
@@ -1130,9 +1130,7 @@ object desugar {
         Apply(Select(Apply(Ident(nme.StringContext), strs), id), elems)
       case InfixOp(l, op, r) =>
         if (ctx.mode is Mode.Type)
-          if (!op.isBackquoted && op.name == tpnme.raw.AMP) AndTypeTree(l, r)     // l & r
-          else if (!op.isBackquoted && op.name == tpnme.raw.BAR) OrTypeTree(l, r) // l | r
-          else AppliedTypeTree(op, l :: r :: Nil) // op[l, r]
+          AppliedTypeTree(op, l :: r :: Nil) // op[l, r]
         else {
           assert(ctx.mode is Mode.Pattern) // expressions are handled separately by `binop`
           Apply(op, l :: r :: Nil) // op(l, r)
diff --git a/compiler/src/dotty/tools/dotc/parsing/Parsers.scala b/compiler/src/dotty/tools/dotc/parsing/Parsers.scala
index edaf6850d8cf..f3a1218f69aa 100644
--- a/compiler/src/dotty/tools/dotc/parsing/Parsers.scala
+++ b/compiler/src/dotty/tools/dotc/parsing/Parsers.scala
@@ -452,7 +452,7 @@ object Parsers {
       if (isLeftAssoc(op1) != op2LeftAssoc)
         syntaxError(MixedLeftAndRightAssociativeOps(op1, op2, op2LeftAssoc), offset)
 
-    def reduceStack(base: List[OpInfo], top: Tree, prec: Int, leftAssoc: Boolean, op2: Name): Tree = {
+    def reduceStack(base: List[OpInfo], top: Tree, prec: Int, leftAssoc: Boolean, op2: Name, isType: Boolean): Tree = {
       if (opStack != base && precedence(opStack.head.operator.name) == prec)
         checkAssoc(opStack.head.offset, opStack.head.operator.name, op2, leftAssoc)
       def recur(top: Tree): Tree = {
@@ -464,7 +464,15 @@ object Parsers {
             opStack = opStack.tail
             recur {
               atPos(opInfo.operator.pos union opInfo.operand.pos union top.pos) {
-                InfixOp(opInfo.operand, opInfo.operator, top)
+                val op = opInfo.operator
+                val l = opInfo.operand
+                val r = top
+                if (isType && !op.isBackquoted && op.name == tpnme.raw.BAR) {
+                  OrTypeTree(checkAndOrArgument(l), checkAndOrArgument(r))
+                } else if (isType && !op.isBackquoted && op.name == tpnme.raw.AMP) {
+                  AndTypeTree(checkAndOrArgument(l), checkAndOrArgument(r))
+                } else
+                  InfixOp(l, op, r)
               }
             }
           }
@@ -488,20 +496,20 @@ object Parsers {
       var top = first
       while (isIdent && isOperator) {
         val op = if (isType) typeIdent() else termIdent()
-        top = reduceStack(base, top, precedence(op.name), isLeftAssoc(op.name), op.name)
+        top = reduceStack(base, top, precedence(op.name), isLeftAssoc(op.name), op.name, isType)
         opStack = OpInfo(top, op, in.offset) :: opStack
         newLineOptWhenFollowing(canStartOperand)
         if (maybePostfix && !canStartOperand(in.token)) {
           val topInfo = opStack.head
           opStack = opStack.tail
-          val od = reduceStack(base, topInfo.operand, 0, true, in.name)
+          val od = reduceStack(base, topInfo.operand, 0, true, in.name, isType)
           return atPos(startOffset(od), topInfo.offset) {
             PostfixOp(od, topInfo.operator)
           }
         }
         top = operand()
       }
-      reduceStack(base, top, 0, true, in.name)
+      reduceStack(base, top, 0, true, in.name, isType)
     }
 
 /* -------- IDENTIFIERS AND LITERALS ------------------------------------------- */
@@ -709,15 +717,7 @@ object Parsers {
     /** Same as [[typ]], but if this results in a wildcard it emits a syntax error and
      *  returns a tree for type `Any` instead.
      */
-    def toplevelTyp(): Tree = {
-      val t = typ()
-      findWildcardType(t) match {
-        case Some(wildcardPos) =>
-          syntaxError(UnboundWildcardType(), wildcardPos)
-          scalaAny
-        case None => t
-      }
-    }
+    def toplevelTyp(): Tree = checkWildcard(typ())
 
     /** Type        ::=  [FunArgMods] FunArgTypes `=>' Type
      *                |  HkTypeParamClause `->' Type
@@ -768,9 +768,16 @@ object Parsers {
             accept(RPAREN)
             if (imods.is(Implicit) || isValParamList || in.token == ARROW) functionRest(ts)
             else {
-              for (t <- ts)
-                if (t.isInstanceOf[ByNameTypeTree])
-                  syntaxError(ByNameParameterNotSupported())
+              val ts1 =
+                for (t <- ts) yield {
+                  t match {
+                    case t@ByNameTypeTree(t1) =>
+                      syntaxError(ByNameParameterNotSupported(t), t.pos)
+                      t1
+                    case _ =>
+                      t
+                  }
+                }
               val tuple = atPos(start) { makeTupleOrParens(ts) }
               infixTypeRest(
                 refinedTypeRest(
@@ -784,7 +791,7 @@ object Parsers {
           val start = in.offset
           val tparams = typeParamClause(ParamOwner.TypeParam)
           if (in.token == ARROW)
-            atPos(start, in.skipToken())(LambdaTypeTree(tparams, typ()))
+            atPos(start, in.skipToken())(LambdaTypeTree(tparams, toplevelTyp()))
           else { accept(ARROW); typ() }
         }
         else infixType()
@@ -822,7 +829,7 @@ object Parsers {
 
     def refinedTypeRest(t: Tree): Tree = {
       newLineOptWhenFollowedBy(LBRACE)
-      if (in.token == LBRACE) refinedTypeRest(atPos(startOffset(t)) { RefinedTypeTree(t, refinement()) })
+      if (in.token == LBRACE) refinedTypeRest(atPos(startOffset(t)) { RefinedTypeTree(checkWildcard(t), refinement()) })
       else t
     }
 
@@ -835,7 +842,7 @@ object Parsers {
         if (ctx.settings.strict.value)
           deprecationWarning(DeprecatedWithOperator())
         in.nextToken()
-        AndTypeTree(t, withType())
+        AndTypeTree(checkAndOrArgument(t), checkAndOrArgument(withType()))
       }
       else t
 
@@ -886,7 +893,7 @@ object Parsers {
     private def simpleTypeRest(t: Tree): Tree = in.token match {
       case HASH => simpleTypeRest(typeProjection(t))
       case LBRACKET => simpleTypeRest(atPos(startOffset(t)) {
-        AppliedTypeTree(t, typeArgs(namedOK = false, wildOK = true)) })
+        AppliedTypeTree(checkWildcard(t), typeArgs(namedOK = false, wildOK = true)) })
       case _ => t
     }
 
@@ -917,7 +924,7 @@ object Parsers {
           else Nil
         first :: rest
       }
-      def typParser() = if (wildOK) typ() else toplevelTyp()
+      def typParser() = checkWildcard(typ(), wildOK)
       if (namedOK && in.token == IDENTIFIER)
         typParser() match {
           case Ident(name) if in.token == EQUALS =>
@@ -1001,17 +1008,46 @@ object Parsers {
       else if (location == Location.InPattern) refinedType()
       else infixType()
 
-    /** Checks whether `t` is a wildcard type.
-     *  If it is, returns the [[Position]] where the wildcard occurs.
+    /** Checks whether `t` represents a non-value type (wildcard types, or ByNameTypeTree).
+     *  If it is, returns the [[Tree]] which immediately represents the non-value type.
      */
     @tailrec
-    private final def findWildcardType(t: Tree): Option[Position] = t match {
-      case TypeBoundsTree(_, _) => Some(t.pos)
-      case Parens(t1) => findWildcardType(t1)
-      case Annotated(t1, _) => findWildcardType(t1)
+    private final def findNonValueTypeTree(t: Tree, alsoNonValue: Boolean): Option[Tree] = t match {
+      case TypeBoundsTree(_, _) => Some(t)
+      case ByNameTypeTree(_) if alsoNonValue => Some(t)
+      case Parens(t1) => findNonValueTypeTree(t1, alsoNonValue)
+      case Annotated(t1, _) => findNonValueTypeTree(t1, alsoNonValue)
       case _ => None
     }
 
+    def rejectWildcard(t: Tree, fallbackTree: Tree): Tree =
+      findNonValueTypeTree(t, false) match {
+        case Some(wildcardTree) =>
+          syntaxError(UnboundWildcardType(), wildcardTree.pos)
+          fallbackTree
+        case None => t
+      }
+
+
+    def checkWildcard(t: Tree, wildOK: Boolean = false, fallbackTree: Tree = scalaAny): Tree =
+      if (wildOK)
+        t
+      else
+        rejectWildcard(t, fallbackTree)
+
+    def checkAndOrArgument(t: Tree): Tree =
+      findNonValueTypeTree(t, true) match {
+        case Some(typTree) =>
+          typTree match {
+            case typTree: TypeBoundsTree =>
+              syntaxError(UnboundWildcardType(), typTree.pos)
+            case typTree: ByNameTypeTree =>
+              syntaxError(ByNameParameterNotSupported(typTree), typTree.pos)
+          }
+          scalaAny
+        case None => t
+      }
+
 /* ----------- EXPRESSIONS ------------------------------------------------ */
 
     /** EqualsExpr ::= `=' Expr
@@ -2148,7 +2184,7 @@ object Parsers {
         in.token match {
           case EQUALS =>
             in.nextToken()
-            TypeDef(name, lambdaAbstract(tparams, typ())).withMods(mods).setComment(in.getDocComment(start))
+            TypeDef(name, lambdaAbstract(tparams, toplevelTyp())).withMods(mods).setComment(in.getDocComment(start))
           case SUPERTYPE | SUBTYPE | SEMI | NEWLINE | NEWLINES | COMMA | RBRACE | EOF =>
             TypeDef(name, lambdaAbstract(tparams, typeBounds())).withMods(mods).setComment(in.getDocComment(start))
           case _ =>
@@ -2276,7 +2312,8 @@ object Parsers {
     /** ConstrApp         ::=  SimpleType {ParArgumentExprs}
      */
     val constrApp = () => {
-      val t = annotType()
+      // Using Ident(nme.ERROR) to avoid causing cascade errors on non-user-written code
+      val t = checkWildcard(annotType(), fallbackTree = Ident(nme.ERROR))
       if (in.token == LPAREN) parArgumentExprss(wrapNew(t))
       else t
     }
diff --git a/compiler/src/dotty/tools/dotc/reporting/diagnostic/messages.scala b/compiler/src/dotty/tools/dotc/reporting/diagnostic/messages.scala
index 3abc06caddf7..ef685b0dc0f2 100644
--- a/compiler/src/dotty/tools/dotc/reporting/diagnostic/messages.scala
+++ b/compiler/src/dotty/tools/dotc/reporting/diagnostic/messages.scala
@@ -696,10 +696,10 @@ object messages {
     }
   }
 
-  case class ByNameParameterNotSupported()(implicit ctx: Context)
+  case class ByNameParameterNotSupported(tpe: untpd.TypTree)(implicit ctx: Context)
   extends Message(ByNameParameterNotSupportedID) {
     val kind = "Syntax"
-    val msg = "By-name parameter type not allowed here."
+    val msg = hl"By-name parameter type ${tpe} not allowed here."
 
     val explanation =
       hl"""|By-name parameters act like functions that are only evaluated when referenced,
diff --git a/tests/neg/i4373.scala b/tests/neg/i4373.scala
new file mode 100644
index 000000000000..6ba7c81a617a
--- /dev/null
+++ b/tests/neg/i4373.scala
@@ -0,0 +1,51 @@
+trait Base
+trait TypeConstr[X]
+
+class X1[A >: _ | X1[_]] // error
+class X2[A >: _ & X2[_]] // error
+class X3[A >: X3[_] | _] // error
+class X4[A >: X4[_] & _] // error
+class X5[A >: _ with X5[_]] // error
+class X6[A >: X6[_] with _] // error
+
+class A1 extends _ // error
+class A2 extends _ with _ // error // error
+class A3 extends Base with _ // error
+class A4 extends _ with Base // error
+
+object Test {
+  type T1 = _ // error
+  type T2 = _[Int] // error
+  type T3 = _ { type S } // error
+  type T4 = [X] => _ // error
+
+  // Open questions:
+  type T5 = TypeConstr[_ { type S }] // error
+  type T6 = TypeConstr[_[Int]] // error
+
+  // expression types
+  type T7 = (=> Int) | (Int => Int) // error
+  type T8 = (=> Int) & (Int => Int) // error
+  type T9 = (=> Int) with (Int => Int) // error
+  type T10 = (Int => Int) | (=> Int) // error
+  type T11 = (Int => Int) & (=> Int) // error
+  type T12 = (Int => Int) with (=> Int) // error
+
+  // annotations
+  type T13 = _ @ annotation.tailrec // error
+  type T14 = Int @ _ // error
+  type T15 = (_ | Int) @ annotation.tailrec // error
+  type T16 = (Int | _) @ annotation.tailrec // error
+  type T17 = Int @ (_ | annotation.tailrec) // error
+  type T18 = Int @ (annotation.tailrec | _) // error
+
+  type T19 = (_ with Int) @ annotation.tailrec // error
+  type T20 = (Int with _) @ annotation.tailrec // error
+  type T21 = Int @ (_ with annotation.tailrec) // error // error
+  type T22 = Int @ (annotation.tailrec with _) // error // error
+
+  type T23 = (_ & Int) @ annotation.tailrec // error
+  type T24 = (Int & _) @ annotation.tailrec // error
+  type T25 = Int @ (_ & annotation.tailrec) // error
+  type T26 = Int @ (annotation.tailrec & _) // error
+}
diff --git a/tests/neg/i4373a.scala b/tests/neg/i4373a.scala
new file mode 100644
index 000000000000..d0e9505b8c44
--- /dev/null
+++ b/tests/neg/i4373a.scala
@@ -0,0 +1,3 @@
+// ==> 040fb47fbaf718cecb11a7d51ac5a48bf4f6a1fe.scala <==
+object x0 {
+  val x0 : _ with // error // error // error
diff --git a/tests/neg/i4373b.scala b/tests/neg/i4373b.scala
new file mode 100644
index 000000000000..dee55114da66
--- /dev/null
+++ b/tests/neg/i4373b.scala
@@ -0,0 +1,4 @@
+// ==> 05bef7805687ba94da37177f7568e3ba7da1f91c.scala <==
+class x0 {
+  x1: // error
+      x0 | _ // error // error
diff --git a/tests/neg/i4373c.scala b/tests/neg/i4373c.scala
new file mode 100644
index 000000000000..8f0246b6cdee
--- /dev/null
+++ b/tests/neg/i4373c.scala
@@ -0,0 +1,3 @@
+// ==> 18b253a4a89a84c5674165c6fc3efafad535eee3.scala <==
+object x0 {
+  def x1[x2 <:_[ // error // error // error