Abstract: I will give an introduction to the theory of cluster algebras, the heart of which is a change of coordinates (called `mutation') which may be iterated to generate a family of special coordinate systems (called `clusters'). This pattern has been found in the coordinate rings of many notable spaces, such as semisimple Lie groups, Grassmannians, and decorated Teichmueller spaces. It is tempting to hope that all cluster algebras exhibit the same nice algebraic and geometric properties of the preceding examples; ie, regularity, finite generation, etc. However, this is not the case! There are examples of cluster algebras possessing such horrors as a non-Noetherian singularity. In fact, a curious dichotomy has emerged: known examples of cluster algebras are well-behaved or pathological, with a large gap in between. I will survey the theory of `locally acyclic cluster algebras', a sub-class of cluster algebras which eliminates the horrible examples while containing the motivating examples. This includes applications to knot theory via the `Kauffman skein bracket', as well as the enumeration of perfect matchings in planar graphs.