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The FFmpeg project is free and open-source and tries to provide the best technically possible solution for developers of applications and end users alike.
FFmpeg is able to decode, encode, transcode, mux, demux, stream, filter and play pretty much anything. It supports obscure ancient formats up to the cutting edge. No matter if they were designed by some standards committee, the community or a corporation.
SpecsUpdate
Pros
Pro Excellent streaming support
Because ffmpeg allows transcoding on the fly and supports multiple streaming protocols such as rtmp, rtsp, http, ftp, hls, you can use it to stream to services such as twitch.tv or set up your own streaming solution.
You can use both local realtime recordings or another stream as a source, transcode it if necessary, and output it to a different stream.
ffmpeg -i rtmp://server/live/originalStream -c:a copy -c:v libx264 -vpre slow -f flv rtmp://server/live/h264Stream