We provide a new tool for studying cluster algebras by introducing a new category fClus of rooted cluster algebras. We characterize isomorphisms in our new category and show that it is neither complete nor cocomplete. We give a recipe for constructing morphisms in fClus with an interesting geometric interpretation and study the corresponding inverse systems. We define and study a new family of algebras, called pro-cluster algebras, with clusterlike combinatorics. The pro-cluster algebras are generated inside inverse limits of inverse systems in the category fClus. Initially, the generators of a pro-cluster algebra are grouped into certain subsets, called pro-clusters, of an inverse limit. In this new setting pro-clusters take the role of clusters and we construct pro-cluster algebras which are modelled by the combinatorics of infinitely marked surfaces and prove that all triangulations of those surfaces arise as pro-clusters.