Permutation groups are one of the oldest topics in algebra. Their study has recently been revolutionized by new developments, particularly the Classification of Finite Simple Groups, but also relations with logic and combinatorics, and importantly, computer algebra systems have been introduced that can deal with large permutation groups. This text summarizes these developments, including an introduction to relevant computer algebra systems, sketch proofs of major theorems, and many examples of applying the Classification of Finite Simple Groups. It is aimed at beginning graduate students and experts in other areas, and grew from a short course at the EIDMA institute in Eindhoven.