The fundamental group of a space provides essential information about features of the space which are preserved under deformations. Topologists encounter the fundamental group in terms of a presentation by generators and relations. A presentation, typically a finite set of data, uniquely determines the group, but it is notoriously difficult to learn properties of the group from a given presentation. Even the most obvious question “Is the group presented finite or infinite?”, is difficult to answer. In fact, the question is known to be algorithmically undecidable. In my talk I will survey combinatorial and geometric methods that can be used in the study of group presentations. The talk will be suitable for a general audience.