It has been exactly 50 years since Gian-Carlo Rota's classical paper "Foundations of Combinatorial Theory", and since then the field has been revolutionized and vastly expanded. I will attempt to give new answers to these questions by showing the connections to Computer Science and Discrete Geometry, but mostly giving a survey of various recent results in Enumerative Combinatorics. Specific topics include partitions, parking functions, spanning trees, polyomino tiling, integer points in polytopes and (just a little bit of) Young tableaux.
The lectures are aimed at a general combinatorics audience and hopefully will be somewhat entertaining.