diff --git a/repo2docker/semver.py b/repo2docker/semver.py
index 97650fa6..661829ec 100644
--- a/repo2docker/semver.py
+++ b/repo2docker/semver.py
@@ -95,11 +95,32 @@ def create_semver_matcher(constraint_str):
         else:
             return VersionRange(constraint, (major(constraint) + 1,), True)
 
-    # '~' matching (only allowed to bump the last present number by one)
+    # '~' matching, bumps minor version unless only major specified or leading 0.0.
+    # https://pkgdocs.julialang.org/v1/compatibility/#Tilde-specifiers
     if comparison_symbol == "~":
-        return VersionRange(
-            constraint, constraint[:-1] + (constraint[-1] + 1,), exclusive=False
-        )
+        if len(constraint) == 1:
+            # only major specified
+            # ~x -> [x-x+1)
+            return VersionRange(constraint, (constraint[0] + 1,), exclusive=True)
+        elif len(constraint) == 2:
+            # ~x.y -> [x.y-x.y+1)
+            return VersionRange(
+                constraint,
+                (
+                    major(constraint),
+                    minor(constraint) + 1,
+                ),
+                exclusive=True,
+            )
+        else:
+            # 3 parts, different depending on if starts with 0.0
+            if major(constraint) == 0 and minor(constraint) == 0:
+                # ~0.0.3 = [0.0.3-0.0.4)
+                upper = (0, 0, patch(constraint) + 1)
+            else:
+                # ~0.2.3 = [0.2.3-0.3)
+                upper = (major(constraint), minor(constraint) + 1)
+            return VersionRange(constraint, upper, exclusive=True)
 
     # Use semver package's comparisons for everything else:
 
diff --git a/tests/unit/test_semver.py b/tests/unit/test_semver.py
index b07272be..dedbcc2e 100644
--- a/tests/unit/test_semver.py
+++ b/tests/unit/test_semver.py
@@ -29,13 +29,14 @@ def test_major_minor_patch():
         ("^0.0.3", "[0.0.3-0.0.4)"),
         ("^0.0", "[0.0-0.1)"),
         ("^0", "[0-1)"),
-        # This one seems wrong: `~1.2.3 = [1.2.3, 1.2.4)`
-        # but ~ is special in Julia from
-        # https://docs.julialang.org/en/latest/stdlib/Pkg/#Tilde-specifiers-1
-        ("~1.2.3", "[1.2.3-1.2.4]"),
-        ("~1.3.5", "[1.3.5-1.3.6]"),
-        ("~1.2", "[1.2-1.3]"),
-        ("~1", "[1-2]"),
+        # https://pkgdocs.julialang.org/v1/compatibility/#Tilde-specifiers
+        ("~1.2.3", "[1.2.3-1.3)"),
+        ("~1.3.5", "[1.3.5-1.4)"),
+        ("~1.2", "[1.2-1.3)"),
+        ("~1", "[1-2)"),
+        ("~0.0.3", "[0.0.3-0.0.4)"),
+        ("~0.0", "[0.0-0.1)"),
+        ("~0", "[0-1)"),
     ],
 )
 def test_simple_matches(test_input, expected):
@@ -65,6 +66,9 @@ def test_range_matches(test_input, expected):
         ("1.2.3", "1.2.3"),
         ("~1.2.3", "1.2.4"),
         ("~1.2.3", "1.2.3"),
+        ("~1.2", "1.2.10"),
+        ("~1", "1.99"),
+        ("~0.0.3", "0.0.3"),
         ("1.2", "1.2.0"),
         ("1.2", "1.9.9"),
         ("0.2.3", "0.2.3"),
@@ -86,6 +90,9 @@ def test_match_particular_version_expected_true(test_input, expected):
         ("1.2.3", "2.0.0"),
         ("1.2.3", "1.2.2"),
         ("~1.2.3", "1.3"),
+        ("~1.2", "1.3"),
+        ("~1", "2"),
+        ("~0.0.3", "0.0.4"),
         ("1.2", "2.0.0"),
         ("1.2", "1.1.9"),
         ("0.2.3", "0.3.0"),