Really small semigroups

This talk is essentially a summary of my PhD thesis, which is concerned with the classification and enumeration of finite semigroups. The talk has two main parts. In the first part I present theoretical results on the classification of nilpotent semigroups, which form the vast majority of finite semigroups. In the second part I explain the computer search for the remaining semigroups. I utilised GAP and an interface to the constraint solver Minion to enumerate the remaining types of semigroups of order 9. Also, all types of semigroups of orders 1 to 8 were made available as a data library in GAP. A short demonstration of the library Smallsemi is included in the talk.