finite state automata are turing machines with fixed finite bounds on resource use. automata lend themselves well to real-time computations and efficient algorithms. therefore, there is a long tradition of studying that part of mathematics which can be represented by automata. this talk will give a survey of this research. in particular, we discuss three major themes: how complicated can automatic structures be? can they be naturally described? how efficient are the associated algorithms? examples include thurston's automatic groups associated with 3-manifolds and automatic structures associated to model checking and program verification in computer science.