Here is a set of introductory notes on inverse problems, including exercises! WARNING: They're far from comprehensive.
Here are the pdf versions of four talks I gave that the U.S. Air Force Academy, a short course on inverse problems (the notes above go with these). Talk 1 Talk 2 Talk 3 Talk 4
Here are some matlab files that let you play around with the Radon transform/CT scans.
Not all inverse problems, but here are the slides for the four talks I gave during the 2010-2011 academic year at the Air Force Academy:
A talk on how Google works.
A talk on how JPEG Compression Works.
A talk on wavelets.
A very introductory talk on inverse problems.