It is well known that topological σ-models in two dimensions constitute a path-integral approach to the study of holomorphic maps from a Riemann surface Σ to an almost complex manifold K , the most interesting case being that were K is a Kähler manifold. We show that, in the same way, topological σ-models in four dimensions introduce a path-integral approach to the study of triholomorphic maps q: M → N between a four-dimensional riemannian manifold M and an almost quaternionic manifold N . The most interesting cases are those where M, N are hyper-Kähler or quaternionic Kähler. BRST-cohomology translates into intersection theory in the moduli-space of this new class of instantonic maps, that are named hyperinstantons by us. The definition of triholomorphicity that we propose is expressed by the equation q ∗ − J u ∘ q ∗ ∘ j u = 0 , where { j u , u = 1,2,3} is an almost quaternionic structure on M and { J u , u = 1,2,3} is an almost quaternionic structure on N . This is a generalization of the Cauchy-Fueter equations. For M, N hyper-Kähler, this generalization naturally arises by obtaining the topological σ-model as a twisted version of the N = 2 globally supersymmetric σ-model. We discuss various examples of hyperinstantons, in particular on the torus and the K3 surface. We also analyze the coupling of the topological σ-model to topological gravity. The classification of triholomorphic maps and the analysis of their moduli-space is a new and fully open mathematical problem that we believe deserves the attention of both mathematicians and physicists.