You can download RTMP streams with freeware StreamTransport, CooJah + Blader, shareware Replay Media Catcher,...

And you can capture RTMP streams in real time while you watch them using freeware CooJah (without Blader), Orbit Grab Pro, shareware Replay Media Catcher 4.

If you use Mac OS X, try Jaksta for Mac.

If you use Linux, try rtmpsrv and rtmpsuck.

Related:
FAQ: How to record/download/capture/save Adobe Flash .FLV video RTMP stream from embedded player
How to record rtmp:// flash video .flv stream from embedded Adobe Flash Player