diff --git a/compiler/src/dotty/tools/dotc/ast/Desugar.scala b/compiler/src/dotty/tools/dotc/ast/Desugar.scala
index 91627864ac64..121806dd36ed 100644
--- a/compiler/src/dotty/tools/dotc/ast/Desugar.scala
+++ b/compiler/src/dotty/tools/dotc/ast/Desugar.scala
@@ -39,6 +39,11 @@ object desugar {
    */
   val CheckIrrefutable: Property.Key[MatchCheck] = Property.StickyKey()
 
+  /** A multi-line infix operation with the infix operator starting a new line.
+   *  Used for explaining potential errors.
+   */
+  val MultiLineInfix: Property.Key[Unit] = Property.StickyKey()
+
   /** What static check should be applied to a Match? */
   enum MatchCheck {
     case None, Exhaustive, IrrefutablePatDef, IrrefutableGenFrom
@@ -1194,7 +1199,10 @@ object desugar {
         case Tuple(args) => args.mapConserve(assignToNamedArg)
         case _ => arg :: Nil
       }
-      Apply(Select(fn, op.name).withSpan(selectPos), args)
+      val sel = Select(fn, op.name).withSpan(selectPos)
+      if (left.sourcePos.endLine < op.sourcePos.startLine)
+        sel.pushAttachment(MultiLineInfix, ())
+      Apply(sel, args)
     }
 
     if (isLeftAssoc(op.name))
diff --git a/compiler/src/dotty/tools/dotc/parsing/Scanners.scala b/compiler/src/dotty/tools/dotc/parsing/Scanners.scala
index 78bed73dce3a..6181c5ac7c44 100644
--- a/compiler/src/dotty/tools/dotc/parsing/Scanners.scala
+++ b/compiler/src/dotty/tools/dotc/parsing/Scanners.scala
@@ -219,6 +219,9 @@ object Scanners {
   class Scanner(source: SourceFile, override val startFrom: Offset = 0)(implicit ctx: Context) extends ScannerCommon(source)(ctx) {
     val keepComments: Boolean = !ctx.settings.YdropComments.value
 
+    /** A switch whether operators at the start of lines can be infix operators */
+    private var allowLeadingInfixOperators = true
+
     /** All doc comments kept by their end position in a `Map` */
     private[this] var docstringMap: SortedMap[Int, Comment] = SortedMap.empty
 
@@ -265,12 +268,12 @@ object Scanners {
       else IDENTIFIER
     }
 
-    private class TokenData0 extends TokenData
+    def newTokenData: TokenData = new TokenData {}
 
     /** We need one token lookahead and one token history
      */
-    val next : TokenData = new TokenData0
-    private val prev : TokenData = new TokenData0
+    val next = newTokenData
+    private val prev = newTokenData
 
     /** a stack of tokens which indicates whether line-ends can be statement separators
      *  also used for keeping track of nesting levels.
@@ -378,6 +381,30 @@ object Scanners {
         next.token = EMPTY
       }
 
+      def insertNL(nl: Token): Unit = {
+        next.copyFrom(this)
+        //  todo: make offset line-end of previous line?
+        offset = if (lineStartOffset <= offset) lineStartOffset else lastLineStartOffset
+        token = nl
+      }
+
+
+      /** A leading symbolic or backquoted identifier is treated as an infix operator
+       *  if it is followed by at least one ' ' and a token on the same line
+       *  that can start an expression.
+       */
+      def isLeadingInfixOperator =
+        allowLeadingInfixOperators &&
+        (token == BACKQUOTED_IDENT ||
+         token == IDENTIFIER && isOperatorPart(name(name.length - 1))) &&
+        (ch == ' ') && {
+          val lookahead = lookaheadScanner
+          lookahead.allowLeadingInfixOperators = false
+            // force a NEWLINE a after current token if it is on its own line
+          lookahead.nextToken()
+          canStartExpressionTokens.contains(lookahead.token)
+        }
+
       /** Insert NEWLINE or NEWLINES if
        *  - we are after a newline
        *  - we are within a { ... } or on toplevel (wrt sepRegions)
@@ -389,10 +416,15 @@ object Scanners {
           (canStartStatTokens contains token) &&
           (sepRegions.isEmpty || sepRegions.head == RBRACE ||
            sepRegions.head == ARROW && token == CASE)) {
-        next copyFrom this
-        //  todo: make offset line-end of previous line?
-        offset = if (lineStartOffset <= offset) lineStartOffset else lastLineStartOffset
-        token = if (pastBlankLine()) NEWLINES else NEWLINE
+        if (pastBlankLine())
+          insertNL(NEWLINES)
+        else if (!isLeadingInfixOperator)
+          insertNL(NEWLINE)
+        else if (isScala2Mode)
+          ctx.warning(em"""Line starts with an operator;
+                          |it is now treated as a continuation of the expression on the previous line,
+                          |not as a separate statement.""",
+                      source.atSpan(Span(offset)))
       }
 
       postProcessToken()
@@ -1087,8 +1119,6 @@ object Scanners {
       case _ => showToken(token)
     }
 
-// (does not seem to be needed) def flush = { charOffset = offset; nextChar(); this }
-
     /* Resume normal scanning after XML */
     def resume(lastToken: Token): Unit = {
       token = lastToken
diff --git a/compiler/src/dotty/tools/dotc/typer/TypeAssigner.scala b/compiler/src/dotty/tools/dotc/typer/TypeAssigner.scala
index b36a6a7097aa..b08f79904337 100644
--- a/compiler/src/dotty/tools/dotc/typer/TypeAssigner.scala
+++ b/compiler/src/dotty/tools/dotc/typer/TypeAssigner.scala
@@ -271,7 +271,13 @@ trait TypeAssigner {
               |An extension method was tried, but could not be fully constructed:
               |
               |    ${failure.tree.show.replace("\n", "\n    ")}"""
-          case _ => ""
+          case _ =>
+            if (tree.hasAttachment(desugar.MultiLineInfix))
+              i""".
+                 |Note that `$name` is treated as an infix operator in Scala 3.
+                 |If you do not want that, insert a `;` or empty line in front
+                 |or drop any spaces behind the operator."""
+            else ""
         }
       errorType(NotAMember(qualType, name, kind, addendum), tree.sourcePos)
     }
diff --git a/tests/neg/multiLineOps.scala b/tests/neg/multiLineOps.scala
new file mode 100644
index 000000000000..4f48c18af3ce
--- /dev/null
+++ b/tests/neg/multiLineOps.scala
@@ -0,0 +1,15 @@
+val x = 1
+  + 2
+  +3 // error: Expected a toplevel definition
+
+val b1 = {
+  22
+  * 22  // ok
+  */*one more*/22 // error: end of statement expected
+}                 // error: ';' expected, but '}' found
+
+val b2: Boolean = {
+  println(x)
+  ! "hello".isEmpty  // error: value ! is not a member of Unit
+}
+
diff --git a/tests/pos/multiLineOps.scala b/tests/pos/multiLineOps.scala
new file mode 100644
index 000000000000..84af353f6f6b
--- /dev/null
+++ b/tests/pos/multiLineOps.scala
@@ -0,0 +1,26 @@
+val x = 1
+  + 2
+  + 3
+
+class Channel {
+  def ! (msg: String): Channel = this
+  def send_! (msg: String): Channel = this
+}
+
+val c = Channel()
+
+def send() =
+  c ! "hello"
+    ! "world"
+    send_! "!"
+
+val b: Boolean =
+  "hello".isEmpty
+  && true &&
+  !"hello".isEmpty
+
+val b2: Boolean = {
+  println(x)
+  !"hello".isEmpty
+  ???
+}