Amaze File Manager
-free with optional in-app purchases.
-features Material Design, SMB file sharing, a built-in app manager to uninstall apps, root explorer.

Astro File Manager
-free (no adds).
-features include SD card support, cloud storage support, file compression, app management, and archive extraction support (ZIP and RAR).

ASUS File Manager
-free but may contain adds.
-OEM app but compatible with most devices.
-LAN and SMB support, cloud storage support, support for various types of files, archiving support.
-no root access.

FX File Manager
--free/trial versions and premium versions.
-multi-window support, support for network storage like FTP, and support for encrypted archived files. The included text editor and its archive support works with less popular stuff like GZip, Bzip2, and 7zip.
-no ads and it doesnít track your devices. Itís a good choice for those switching away from ES File Explorer.

MiXplorer Silver
-trial and premium versions.
-file browsing, archive support, support for cloud storage, and an HTML viewer, EPub, MobiPlacket, and PDF reader support as well as file encryption. Added plugins for more functionality as needed.

MK Explorer
-free/trial versions and premium versions.
-features a Material Design interface, the basic file management features (copy, paste, delete, SD Card support for Lollipop 5.0+), and root access. There is also support for 20 languages and it has a built-in text editor, gallery, and music player.
-recommend this one for devices running older versions of Android, like Lollipop or Marshmallow.

Root Explorer
-free/trial versions and premium versions.
-frequent updates.
-unmitigated root access (as long as your device is rooted).
-a non-rooted free version is available and is simply labeled Explorer.
-includes tools like an APK binary XML viewer, the ability to change a fileís owner or group, MD5 support, and more.

Solid Explorer
-trial and premium versions.
-features Material Design, cloud support, root access, FTP, SFTP, WebDav, SMB, and CIFS support, archive and compression support, Chromecast support, and with theming options available.

Total Commander
-free with no adds and no in app purchases.
-one of the more powerful and highly rated file manager apps.
-features including network storage support, cloud storage support, plugin support, bookmarks, and other tools such as an included text editor.

X-plore File Manager
-free with in-app purchases.
-Xone of the more unique options on the list. Itís a forced dual-pane app which means youíll be managing two windows at once pretty much all the time. This is kind of cool if youíre copy/pasting between folders or need to move files quickly. It also comes with support for various types of files, cloud storage, network storage (FTP, SMB, DLNA/UPnP), a built-in hex editor, root support, and plenty of other features. You can even view APK files as zips.

thnx to
Joe Hindy