diff --git a/.travis.yml b/.travis.yml
index 37cdad3..3384f24 100644
--- a/.travis.yml
+++ b/.travis.yml
@@ -5,11 +5,20 @@ env:
   matrix:
     - ENV=test
     - ENV=check
+    - ENV=test_nvim
 install:
   - |
     if [ "$ENV" = "test" ]; then
       python3.6 -m pip install pytest
     fi
+  - |
+    # https://github.com/neovim/bot-ci#generated-builds
+    if [ "$ENV" = "test_nvim" ]; then
+      eval "$(curl -Ss https://raw.githubusercontent.com/neovim/bot-ci/master/scripts/travis-setup.sh) nightly-x64"
+      pip --version
+      pip install neovim
+    else
+      vim --version
+    fi
 script:
-  - vim --version
   - make --keep-going "$ENV"
diff --git a/Makefile b/Makefile
index ed049c6..bae7bc4 100644
--- a/Makefile
+++ b/Makefile
@@ -1,6 +1,9 @@
 test:
 	pytest test_*.py
 
+test_nvim:
+	VSPEC_VIM=nvim pytest test_*.py
+
 build:
 	mkdir $@
 
diff --git a/test/completions.vim b/test/completions.vim
index aa6db08..8701d8a 100644
--- a/test/completions.vim
+++ b/test/completions.vim
@@ -13,6 +13,20 @@ describe 'completions'
         bd!
     end
 
+    it 'longest in completeopt'
+        " This gets set up with Vim only on VimEnter.
+        if has('nvim')
+            Expect stridx(&completeopt, 'longest') > -1
+        else
+            Expect stridx(&completeopt, 'longest') == -1
+            doautocmd VimEnter
+            Expect stridx(&completeopt, 'longest') > -1
+        endif
+
+        " Do not use it for following tests.
+        set completeopt-=longest
+    end
+
     it 'smart import'
         exec "normal ifrom os "
         Expect getline('.') == 'from os import '
@@ -35,20 +49,35 @@ describe 'completions'
     it 'exception'
         normal oIndentationErrX
         Expect getline('.') == 'IndentationError'
-        normal a().filenaX
+
+        " Do not remap keys (".") here, otherwise this triggers completion in
+        " Neovim already.
+        normal! a().filena
+
+        normal aX
         Expect getline('.') == 'IndentationError().filename'
     end
 
     it 'multi complete'
+        " NOTE: nvim results in "importErr()" here with completeopt+=longest,
+        " but Vim is fine.
+        " This is due to `pumvisible()` in jedi#complete_opened being true
+        " with nvim still, but it is 0 with Vim, i.e. Vim appears to close
+        " the pum already (with the tests).
+        "
+        " This might be a misunderstanding though, since the test might not
+        " expect the "import" keyword to be offered for completion?!
         normal oImpXErrX()
         Expect getline('.') == 'ImportError()'
     end
 
     it 'cycling through entries popup_select_first=0'
+        set completeopt+=longest
         let g:jedi#popup_select_first = 0
         execute "normal oraise impX\<C-n>"
-        " It looks like this is currently not working properly.
-        "Expect getline('.') == 'raise ImportError'
+
+        Expect getline('.') == 'raise ImportError'
+        set completeopt-=longest
     end
 
     it 'cycling through entries popup_select_first=1'
@@ -56,12 +85,27 @@ describe 'completions'
         Expect getline('.') == 'raise ImportWarning'
     end
 
-    it 'longest'
-        " -longest completes the first one
-        set completeopt -=longest
-        execute "normal oraise baseX"
-        Expect getline('.') == 'raise BaseException'
-        set completeopt +=longest
+    it 'cycling through entries popup_select_first=1 and longest'
+        set completeopt+=longest
+        execute "normal oraise impX"
+        Expect getline('.') == 'raise Import'
+
+        " With Neovim pumvisible() is 1 in jedi#complete_opened, which then
+        " triggers the <Down>.  This is not the case with Vim.
+        if has('nvim')
+            execute "normal oraise impX\<C-n>"
+            Expect getline('.') == 'raise ImportWarning'
+
+            execute "normal oraise impX\<C-n>\<C-n>"
+            Expect getline('.') == 'raise imp'
+        else
+            execute "normal oraise impX\<C-n>"
+            Expect getline('.') == 'raise ImportError'
+
+            execute "normal oraise impX\<C-n>\<C-n>"
+            Expect getline('.') == 'raise ImportWarning'
+        endif
+        set completeopt-=longest
     end
 end
 
diff --git a/test/signatures.vim b/test/signatures.vim
index 9b8a3fc..06533f2 100644
--- a/test/signatures.vim
+++ b/test/signatures.vim
@@ -94,10 +94,10 @@ describe 'signatures'
             return msg
         endfunction
 
-        let funcname = repeat('a', &columns - 30)
+        let funcname = repeat('a', &columns - (30 + (&ruler ? 18 : 0)))
         put = 'def '.funcname.'(arg1, arg2, arg3, a, b, c):'
         put = '    pass'
-        execute "normal o".funcname."( "
+        execute "normal o\<BS>".funcname."( "
         Expect Signature() == "\n".funcname."(arg1, …)"
 
         exe 'normal sarg1, '
@@ -112,7 +112,7 @@ describe 'signatures'
         g/^/d
         put = 'def '.funcname.'('.repeat('b', 20).', arg2):'
         put = '    pass'
-        execute "normal o".funcname."( "
+        execute "normal o\<BS>".funcname."( "
         Expect Signature() == "\n".funcname."(…)"
     end