Cluster algebras, invented by S. Fomin and A. Zelevinsky [J. Am. Math. Soc. 15, No. 2, 497–529 (2002; Zbl 1021.16017)] in order to study total positivity in algebraic groups and canonical bases in quantum groups, is subring of the field \({\mathbb{Q}}(x_1, \dots,x_n)\) of rational functions in \(n\) indeterminates endowed with a distinguished set of generators called cluster variables, which are constructed recursively through mutation. In order to get a better comprehension of cluster algebras, the authors set up a categorical framework for their study. The authors define the {category of rooted cluster algebras} \(\text{Clus}\) whose objects are the rooted cluster algebras and whose morphisms are the rooted cluster morphisms between two rooted cluster algebras which commute with mutations. This category has countable coproducts but has no products in general and in this category, the isomorphisms are the bijective morphisms, the monomorphisms are the injective monomorphisms while the epimorphisms are not necessarily surjective. In the case of cluster algebras from surfaces, the authors described interactions between this category and the geometry of the surfaces. They define for arbitrary cluster algebras concepts of {gluings} and {cuttings} which provide natural classes of monomorphisms and epimorphisms in \(\text{Clus}\). The authors also prove that the usual specialisations of {frozen} variables to 1 yield epimorphisms in \(\text{Clus}\) and more surprisingly, for cluster algebras from surfaces or for acyclic cluster algebras, specialisations of exchangeable cluster variables also give rise to epimorphisms in \(\text{Clus}\).