Live video streaming is still a touchy subject and not well supported across browsers and mobile devices. Hence I did a short overview of how to start on this years WebCamp Ljubljana. The lecture has since been published as a video as well.
Relevant links from the slides:
Pre-built FFMpeg downloads - Official FFMpeg site with precompiled ffmpeg builds for major platforms
Nginx RTMP module - Video streaming server module for Nginx with RTMP, (Apple) HLS and DASH support
Video.js HTML5 video player - Free HTML5 video player that automatically detects browser capabilities and switches to Flash when necessary