diff --git a/cssselect/xpath.py b/cssselect/xpath.py
index 96eac3f..8efd222 100644
--- a/cssselect/xpath.py
+++ b/cssselect/xpath.py
@@ -622,18 +622,30 @@ def xpath_nth_last_child_function(
     ) -> XPathExpr:
         return self.xpath_nth_child_function(xpath, function, last=True)
 
+    @staticmethod
+    def _check_of_type_element(xpath: XPathExpr, pseudo: str) -> None:
+        """Raise an exception if an -of-type pseudo-class can't be used with
+        the given element.
+
+        For "*" and namespace wildcards like "ns:*" the type of the element is
+        not known, and counting same-type siblings cannot be expressed as an
+        XPath 1.0 node test.
+        """
+        element = xpath.element
+        if element == "*" or element.endswith(":*"):
+            css_element = element.replace(":", "|")
+            raise ExpressionError(f"{css_element}:{pseudo} is not implemented")
+
     def xpath_nth_of_type_function(
         self, xpath: XPathExpr, function: Function
     ) -> XPathExpr:
-        if xpath.element == "*":
-            raise ExpressionError("*:nth-of-type() is not implemented")
+        self._check_of_type_element(xpath, "nth-of-type()")
         return self.xpath_nth_child_function(xpath, function, add_name_test=False)
 
     def xpath_nth_last_of_type_function(
         self, xpath: XPathExpr, function: Function
     ) -> XPathExpr:
-        if xpath.element == "*":
-            raise ExpressionError("*:nth-of-type() is not implemented")
+        self._check_of_type_element(xpath, "nth-last-of-type()")
         return self.xpath_nth_child_function(
             xpath, function, last=True, add_name_test=False
         )
@@ -678,22 +690,26 @@ def xpath_last_child_pseudo(self, xpath: XPathExpr) -> XPathExpr:
         return xpath.add_condition("count(following-sibling::*) = 0")
 
     def xpath_first_of_type_pseudo(self, xpath: XPathExpr) -> XPathExpr:
-        if xpath.element == "*":
-            raise ExpressionError("*:first-of-type is not implemented")
+        self._check_of_type_element(xpath, "first-of-type")
         return xpath.add_condition(f"count(preceding-sibling::{xpath.element}) = 0")
 
     def xpath_last_of_type_pseudo(self, xpath: XPathExpr) -> XPathExpr:
-        if xpath.element == "*":
-            raise ExpressionError("*:last-of-type is not implemented")
+        self._check_of_type_element(xpath, "last-of-type")
         return xpath.add_condition(f"count(following-sibling::{xpath.element}) = 0")
 
     def xpath_only_child_pseudo(self, xpath: XPathExpr) -> XPathExpr:
-        return xpath.add_condition("count(parent::*/child::*) = 1")
+        # Count siblings, not the parent's children: the root element has
+        # no parent, but it has no siblings either, so it must match.
+        return xpath.add_condition(
+            "count(preceding-sibling::*) = 0 and count(following-sibling::*) = 0"
+        )
 
     def xpath_only_of_type_pseudo(self, xpath: XPathExpr) -> XPathExpr:
-        if xpath.element == "*":
-            raise ExpressionError("*:only-of-type is not implemented")
-        return xpath.add_condition(f"count(parent::*/child::{xpath.element}) = 1")
+        self._check_of_type_element(xpath, "only-of-type")
+        return xpath.add_condition(
+            f"count(preceding-sibling::{xpath.element}) = 0 "
+            f"and count(following-sibling::{xpath.element}) = 0"
+        )
 
     def xpath_empty_pseudo(self, xpath: XPathExpr) -> XPathExpr:
         return xpath.add_condition("not(*) and not(string-length())")
diff --git a/tests/test_cssselect.py b/tests/test_cssselect.py
index dc67bb7..299b134 100644
--- a/tests/test_cssselect.py
+++ b/tests/test_cssselect.py
@@ -515,8 +515,12 @@ def xpath(css: str) -> str:
         assert xpath("e:last-child") == ("e[count(following-sibling::*) = 0]")
         assert xpath("e:first-of-type") == ("e[count(preceding-sibling::e) = 0]")
         assert xpath("e:last-of-type") == ("e[count(following-sibling::e) = 0]")
-        assert xpath("e:only-child") == ("e[count(parent::*/child::*) = 1]")
-        assert xpath("e:only-of-type") == ("e[count(parent::*/child::e) = 1]")
+        assert xpath("e:only-child") == (
+            "e[count(preceding-sibling::*) = 0 and count(following-sibling::*) = 0]"
+        )
+        assert xpath("e:only-of-type") == (
+            "e[count(preceding-sibling::e) = 0 and count(following-sibling::e) = 0]"
+        )
         assert xpath("e:empty") == ("e[not(*) and not(string-length())]")
         assert xpath("e:EmPTY") == ("e[not(*) and not(string-length())]")
         assert xpath("e:root") == ("e[not(parent::*)]")
@@ -576,6 +580,16 @@ def xpath(css: str) -> str:
             xpath(":nth-of-type(1)")
         with pytest.raises(ExpressionError):
             xpath(":nth-last-of-type(1)")
+        with pytest.raises(ExpressionError):
+            xpath("ns|*:first-of-type")
+        with pytest.raises(ExpressionError):
+            xpath("ns|*:only-of-type")
+        with pytest.raises(ExpressionError):
+            xpath("ns|*:last-of-type")
+        with pytest.raises(ExpressionError):
+            xpath("ns|*:nth-of-type(1)")
+        with pytest.raises(ExpressionError):
+            xpath("ns|*:nth-last-of-type(1)")
         with pytest.raises(ExpressionError):
             xpath(":nth-child(n-)")
         with pytest.raises(ExpressionError):
@@ -977,9 +991,13 @@ def pcss(main: str, *selectors: str, **kwargs: bool) -> list[str]:
         assert pcss("span:only-child") == ["foobar-span"]
         assert pcss("li div:only-child") == ["li-div"]
         assert pcss("div *:only-child") == ["li-div", "foobar-span"]
+        # The root element has no siblings, so it matches :only-child
+        # (just like :first-child and :last-child)
+        assert pcss("html:only-child") == ["html"]
         with pytest.raises(ExpressionError):
             pcss("p *:only-of-type")
         assert pcss("p:only-of-type") == ["paragraph"]
+        assert pcss("html:only-of-type") == ["html"]
         assert pcss("a:empty", "a:EMpty") == ["name-anchor"]
         assert pcss("li:empty") == ["third-li", "fourth-li", "fifth-li", "sixth-li"]
         assert pcss(":root", "html:root") == ["html"]