Kodi (XBMC) media center software can be run on Android-based mobile phone
Kodi (formerly XBMC) media center software is often seen on big screens. However it's possible to run Kodi on mobile phone as well. Try nightly builds (at your own risk) which you can find at: http://mirrors.xbmc.org/nightlies/android/arm/ .
My phone is LG Nexus 4 with Android 5 (Lollipop). Kodi .apk file installed just fine after external sources were accepted (see Settings). In order to make Kodi play media files, read through kodi.wiki.