In this talk I will give an overview of the fundamental algorithms for computing semigroups and monoids defined by a finite presentation $\langle A | R \rangle$. Many (most?) natural questions (such as finiteness, triviality, and so on) about finitely presented semigroups are undecidable in general. On the other hand, the available computation methods can still be used to resolve such questions, and are successful in many examples that arise in the specific situations that interest pure mathematicians. In addition surveying various theoretic aspects, I will discuss the different implementations available.