Just to add to the chorus of fans of Downie…
I use Downie regularly to download audio, videos and even pdf’s at times. Having used numerous browser extensions in Firefox in the past, I have to say Downie is par excellence!
Downie current supports 759 sources, and offers a “User-Guided Extraction” for other sources. It also includes a “Login Browser” for sites that require a login to view their audio or video content
I’ve had many email exchanges with Charlie Monroe (developer) and not only does he respond promptly to ‘problem reports’ but is open to suggestions for additional/improved features. Like others have mentioned he regularly updates the app in response to the ever-changing ways sources dish out their media as well as from the ‘direct’ feedback he gets from his children 
And, BTW, he has a helpful companion app called Permute for converting audio, video, and image files to other formats. For example, when videos are just talking heads (without any other visual presentations, I’ll download the .aac version, then convert that to .mp3 to add to my iTunes library.
Downie also, optionally, downloads transcripts (.srt). I find that extremely useful when searching my ‘information database’ for information of interest contained within audio & visual files.
Charlie Monroe Software: