diff --git a/fsspec/asyn.py b/fsspec/asyn.py
index e6d5edb9d..496b81188 100644
--- a/fsspec/asyn.py
+++ b/fsspec/asyn.py
@@ -703,25 +703,24 @@ async def _walk(self, path, maxdepth=None, on_error="omit", **kwargs):
             ):
                 yield _
 
-    async def _glob(self, path, **kwargs):
+    async def _glob(self, path, maxdepth=None, **kwargs):
+        if maxdepth is not None and maxdepth < 1:
+            raise ValueError("maxdepth must be at least 1")
+
         import re
 
         ends = path.endswith("/")
         path = self._strip_protocol(path)
-        indstar = path.find("*") if path.find("*") >= 0 else len(path)
-        indques = path.find("?") if path.find("?") >= 0 else len(path)
-        indbrace = path.find("[") if path.find("[") >= 0 else len(path)
+        idx_star = path.find("*") if path.find("*") >= 0 else len(path)
+        idx_qmark = path.find("?") if path.find("?") >= 0 else len(path)
+        idx_brace = path.find("[") if path.find("[") >= 0 else len(path)
 
-        ind = min(indstar, indques, indbrace)
+        min_idx = min(idx_star, idx_qmark, idx_brace)
 
         detail = kwargs.pop("detail", False)
 
         if not has_magic(path):
-            root = path
-            depth = 1
-            if ends:
-                path += "/*"
-            elif await self._exists(path):
+            if await self._exists(path):
                 if not detail:
                     return [path]
                 else:
@@ -731,13 +730,21 @@ async def _glob(self, path, **kwargs):
                     return []  # glob of non-existent returns empty
                 else:
                     return {}
-        elif "/" in path[:ind]:
-            ind2 = path[:ind].rindex("/")
-            root = path[: ind2 + 1]
-            depth = None if "**" in path else path[ind2 + 1 :].count("/") + 1
+        elif "/" in path[:min_idx]:
+            min_idx = path[:min_idx].rindex("/")
+            root = path[: min_idx + 1]
+            depth = path[min_idx + 1 :].count("/") + 1
         else:
             root = ""
-            depth = None if "**" in path else path[ind + 1 :].count("/") + 1
+            depth = path[min_idx + 1 :].count("/") + 1
+
+        if "**" in path:
+            if maxdepth is not None:
+                idx_double_stars = path.find("**")
+                depth_double_stars = path[idx_double_stars:].count("/") + 1
+                depth = depth - depth_double_stars + maxdepth
+            else:
+                depth = None
 
         allpaths = await self._find(
             root, maxdepth=depth, withdirs=True, detail=True, **kwargs
@@ -765,14 +772,23 @@ async def _glob(self, path, **kwargs):
             )
             + "$"
         )
-        pattern = re.sub("[*]{2}", "=PLACEHOLDER=", pattern)
+        pattern = re.sub("/[*]{2}", "=SLASH_DOUBLE_STARS=", pattern)
+        pattern = re.sub("[*]{2}/?", "=DOUBLE_STARS=", pattern)
         pattern = re.sub("[*]", "[^/]*", pattern)
-        pattern = re.compile(pattern.replace("=PLACEHOLDER=", ".*"))
+        pattern = re.sub("=SLASH_DOUBLE_STARS=", "(|/.*)", pattern)
+        pattern = re.sub("=DOUBLE_STARS=", ".*", pattern)
+        pattern = re.compile(pattern)
         out = {
             p: allpaths[p]
             for p in sorted(allpaths)
             if pattern.match(p.replace("//", "/").rstrip("/"))
         }
+
+        # Return directories only when the glob end by a slash
+        # This is needed for posix glob compliance
+        if ends:
+            out = {k: v for k, v in out.items() if v["type"] == "directory"}
+
         if detail:
             return out
         else:
@@ -793,6 +809,12 @@ async def _find(self, path, maxdepth=None, withdirs=False, **kwargs):
         path = self._strip_protocol(path)
         out = dict()
         detail = kwargs.pop("detail", False)
+
+        # Add the root directory if withdirs is requested
+        # This is needed for posix glob compliance
+        if withdirs and await self._isdir(path):
+            out[path] = await self._info(path)
+
         # async for?
         async for _, dirs, files in self._walk(path, maxdepth, detail=True, **kwargs):
             if withdirs:
@@ -819,7 +841,7 @@ async def _expand_path(self, path, recursive=False, maxdepth=None):
             path = [self._strip_protocol(p) for p in path]
             for p in path:  # can gather here
                 if has_magic(p):
-                    bit = set(await self._glob(p))
+                    bit = set(await self._glob(p, maxdepth=maxdepth))
                     out |= bit
                     if recursive:
                         # glob call above expanded one depth so if maxdepth is defined
diff --git a/fsspec/spec.py b/fsspec/spec.py
index 224623960..0eb513032 100644
--- a/fsspec/spec.py
+++ b/fsspec/spec.py
@@ -635,8 +635,12 @@ def glob(self, path, maxdepth=None, **kwargs):
             for p in sorted(allpaths)
             if pattern.match(p.replace("//", "/").rstrip("/"))
         }
+
+        # Return directories only when the glob end by a slash
+        # This is needed for posix glob compliance
         if ends:
             out = {k: v for k, v in out.items() if v["type"] == "directory"}
+
         if detail:
             return out
         else: