A $(k, g)$-cage is a $k$-regular graph of girth $g$ of minimum order. In this survey, we present the results of over 50 years of searches for cages. We present the important theorems, list all the known cages, compile tables of current record holders, and describe in some detail most of the relevant constructions.

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We present a theory of Cayley maps, i.e., embeddings of Cayley graphs into oriented surfaces having the same cyclic rotation of generators around each vertex. These maps have often been used to encode symmetric embeddings of graphs. We also present an algebraic theory of Cayley maps and we apply the theory to determine exactly which regular or edge-transitive tilings of the sphere or plane are Cayley maps or Cayley graphs. Our main goal, however, is to provide the general theory so as to make it easier for others to study Cayley maps.

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The concept of a t-balancedCayley map is a natural generalization of the previously studied notions of balanced and anti-balanced Cayley maps (the terms coined by [J. Širáň, M. Škoviera, Groups with sign structure and their antiautomorphisms, Discrete Math. 108 (1992) 189-202. [12]]). We develop a general theory of t-balanced Cayley maps based on the use of skew-morphisms of groups [R. Jajcay, J. Širáň, Skewmorphisms of regular Cayley maps, Discrete Math. 244 (1-3) (2002) 167-179], and apply our results to the specific case of regular Cayley maps of abelian groups.

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