diff --git a/doc/fugitive.txt b/doc/fugitive.txt
index 6c27c6f..c3a9ffb 100644
--- a/doc/fugitive.txt
+++ b/doc/fugitive.txt
@@ -196,7 +196,13 @@ that are part of Git repositories).
 :Gmove {destination}    Wrapper around git-mv that renames the buffer
                         afterward.  The destination is relative to the current
                         directory except when started with a /, in which case
-                        it is relative to the work tree.  Add a ! to pass -f.
+                        it is relative to the work tree.  (This is a holdover
+                        from before |:Grename| and will be removed.)  Add a !
+                        to pass -f.
+
+                                                *fugitive-:Grename*
+:Grename {destination}  Like |:Gmove| but operates relative to the parent
+                        directory of the current file.
 
                                                 *fugitive-:Gdelete*
 :Gdelete                Wrapper around git-rm that deletes the buffer
diff --git a/plugin/fugitive.vim b/plugin/fugitive.vim
index d543dd3..cdd0563 100644
--- a/plugin/fugitive.vim
+++ b/plugin/fugitive.vim
@@ -1883,9 +1883,11 @@ endfunction
 
 " Section: Gmove, Gremove
 
-function! s:Move(force,destination) abort
+function! s:Move(force, rename, destination) abort
   if a:destination =~# '^/'
     let destination = a:destination[1:-1]
+  elseif a:rename
+    let destination = fnamemodify(s:buffer().path(), ':h') . '/' . a:destination
   else
     let destination = s:shellslash(fnamemodify(s:sub(a:destination,'[%#]%(:\w)*','\=expand(submatch(0))'),':p'))
     if destination[0:strlen(s:repo().tree())] ==# s:repo().tree('')
@@ -1906,7 +1908,7 @@ function! s:Move(force,destination) abort
     let destination = fnamemodify(s:sub(destination,'/$','').'/'.expand('%:t'),':.')
   endif
   call fugitive#reload_status()
-  if s:buffer().commit() == ''
+  if empty(s:buffer().commit())
     if isdirectory(destination)
       return 'keepalt edit '.s:fnameescape(destination)
     else
@@ -1918,15 +1920,24 @@ function! s:Move(force,destination) abort
 endfunction
 
 function! s:MoveComplete(A,L,P) abort
-  if a:A =~ '^/'
+  if a:A =~# '^/'
     return s:repo().superglob(a:A)
   else
     let matches = split(glob(a:A.'*'),"\n")
-    call map(matches,'v:val !~ "/$" && isdirectory(v:val) ? v:val."/" : v:val')
+    call map(matches,'v:val !~# "/$" && isdirectory(v:val) ? v:val."/" : v:val')
     return matches
   endif
 endfunction
 
+function! s:RenameComplete(A,L,P) abort
+  if a:A =~# '^/'
+    return s:repo().superglob(a:A)
+  else
+    let pre = '/'. fnamemodify(s:buffer().path(), ':h') . '/'
+    return map(s:repo().superglob(pre.a:A), 'strpart(v:val, len(pre))')
+  endif
+endfunction
+
 function! s:Remove(after, force) abort
   if s:buffer().commit() ==# ''
     let cmd = ['rm']
@@ -1952,7 +1963,8 @@ endfunction
 augroup fugitive_remove
   autocmd!
   autocmd User Fugitive if s:buffer().commit() =~# '^0\=$' |
-        \ exe "command! -buffer -bar -bang -nargs=1 -complete=customlist,s:MoveComplete Gmove :execute s:Move(<bang>0,<q-args>)" |
+        \ exe "command! -buffer -bar -bang -nargs=1 -complete=customlist,s:MoveComplete Gmove :execute s:Move(<bang>0,0,<q-args>)" |
+        \ exe "command! -buffer -bar -bang -nargs=1 -complete=customlist,s:RenameComplete Grename :execute s:Move(<bang>0,1,<q-args>)" |
         \ exe "command! -buffer -bar -bang Gremove :execute s:Remove('edit',<bang>0)" |
         \ exe "command! -buffer -bar -bang Gdelete :execute s:Remove('bdelete',<bang>0)" |
         \ endif