I think at some point there was a Coursera course in approximation algorithms given by Claire Mathieu, who is one of the top researchers in approximation algorithms. Here's a link https://www.coursera.org/learn/approximation-algorithms-part-1?
In addition to that there is the textbook by Williamson and Shmoys which is available for free online. http://www.designofapproxalgs.com/