In this chapter, we introduce cluster varieties and cluster algebras. Cluster varieties allow a special atlas in which the transition functions are certain birational transformations called mutations. Many moduli spaces associated to ciliated surfaces allow this structure, for example the moduli space of n points in the projective line. On certain spaces associated to ciliated surfaces there exist natural coordinate systems, one for each triangulation of the surface. In this sense, a triangulation provides a chart on such a space, where a chart is to be understood rather in the sense of algebraic geometry than differential geometry since each chart covers an open dense subset of the space. Cluster algebras provide a profound algebraic structure which helps us understand the transition functions between two such charts or, more explicitly, how to relate the coordinate systems corresponding to two distinct triangulations of the same underlying ciliated surface. Cluster algebras are rather recent algebraic structures introduced by Fomin, Zelevinsky and Berenstein in a series of papers (Fomin and Zelevinsky, J Am Math Soc 15(2):497–529;2002. ISSN: 0894-0347. https://doi.org/10.1090/S0894-0347-01-00385-X . arXiv: math/0104151, Fomin and Zelevinsky, Invent Math 154(1):63–121;2003. ISSN: 0020-9910. https://doi.org/10.1007/s00222-003-0302-y . arXiv: math/0208229, Berenstein et al., Duke Math J 126(1), 1–52;2005. ISSN: 0012-7094. https://doi.org/10.1215/S0012-7094-04-12611-9 . arXiv: math/0305434, Fomin and Zelevinsky, Compos Math 143(1),112–164;2007. ISSN: 0010-437X. https://doi.org/10.1112/S0010437X06002521 . arXiv: math/0602259).