Many softwares exist for solving partial differential equations through the finite element method. We propose a new library, written from scratch entirely in C++, called maniFEM (https://manifem.rd.ciencias.ulisboa.pt). The name comes from “finite elements on manifolds". ManiFEM is free software (LGPL). ManiFEM was designed with the goal of coping with very general meshes, in particular meshes on Riemannian manifolds, including quotient manifolds like the flat torus. Also, maniFEM was written with the goal of being conceptually clear and easy to read. We hope it will be particularly useful for people who want fine control over the mesh, e.g. for implementing their own meshing or remeshing algorithms.