Lie groups arise in mathematics and physics as groups of continuous symmetries. The theory of finite-dimensional Lie groups and their representations has a deep and rich theory, and plays an important role in modern physics, for example in quantum mechanics and in gauge theories. However, sometimes a group of symmetries depends on too many parameters to be modelled on a finite-dimensional space! Such spaces are also objects of interest: for example, loop spaces, diffeomorphism groups and groups of gauge transformations. In this talk, we introduce the notion of infinite-dimensional (locally convex) Lie groups, as treated by John Milnor and discuss what familiar features carry over from the finite-dimensional setting. In finite dimensions, there are many powerful tools for obtaining local or global information about a Lie group from its Lie algebra, for example via the exponential map. We discuss the limitations of this approach in infinite dimensions, and describe what partial results can still be obtained in this setting