Okay I have noticed alot of people recently have being asking this question so heres a tut
First of you need to download this_
Free youtube to ipod converter(the ipod converter also converts to psp)
http://www.sendspace.com/file/vr4ifh
1)download and install the converter
2)go to the video you want and copy the URL
3)open the converter and click continue
4)paste the URL into the top text box
5)change device to sony PSP
6)click convert
7)once it finishes converting find the file (default is C drive)
8)copy the file into the video folder of your psp
9)go to videos on the XMB and watch it
