Cluster algebras were introduced by Fomin and Zelevinsky in early 2000 in the context of Lie theory. Cluster algebras are commutative rings with a set of distinguished generators. Due to their rich combinatorial structure, the theory of cluster algebras has spread to many other areas of mathematics, from triangulations of surfaces, to Teichmueller theory, to quiver representations. Fock and Gonchorov, Fomin, Shapiro, and Thurston, and Gekhtman, Shapiro, and Vahnstein established a relation between cluster algebras and hyperbolic geometry. Explicit combinatorial formulas for cluster variables in terms of perfect matchings of snake graphs were given by Musiker, Schiffler, and Williams. Thus the curves in the surface completely determine the combinatorial and algebraic structure of the cluster algebra. In this thesis, we consider asymptotic triangulations, and describe their cluster algebra structure. We start with asymptotic triangulations of the annulus, and consider the flips of arcs in the triangulations, their associated quivers and the mutation rules of these quivers (Chapter 2), and then look at their associated algebra (Chapter 3). Using lambda lengths and laminations, we get a cluster algebra-like structure with principle coefficients for asymptotic triangulations. Triangulations and asymptotic triangulations of the annulus were used to characterize infinite frieze patterns of integers by Baur, Parsons, and Tschabold. In Chapter 4, we give a cluster-algebraic interpretation of these infinite frieze patterns, by constructing an infinite frieze where the entries are Laurent polynomials of generalized arcs in the triangulated surface. Using snake graphs and skein relations, we achieve some algebraic and combinatorial results involving the relationships between entries in the frieze, and give geometric interpretations of known results for infinite friezes with integer entries.