Cellular materials have been engineered in the last years to reproduce the properties characterizing media available in nature such as sponges, wood, bones, honeycombs, thanks also to the promotion of additive processes like 3D printing. From an innovation viewpoint, the request for new lattices has prompted for the proposal of novel methods for the design of materials. With this respect, topology optimization constitutes the reference mathematical methodology, in combination with techniques such as inverse and direct homogenization. In this presentation we focus on the recently proposed microSIMPATY algorithm which merges inverse homogenization with SIMP method for topology optimization, in the presence of a computational mesh which fosters the design phase. Indeed, the employment of an anisotropic adapted mesh to approximate the optimization design problem allows us to deliver layouts characterized by a smooth and sharp void/material interface and by original free-form features. MicroSIMPATY algorithm proved to be a very flexible and efficient tool to set up a workflow for the design of innovative cellular materials matching target requirements at the macro-scale, in the framework of a mono- and a multi-objective optimization, as well as of a multi-physics and multi-scale design process. Goal of the talk is to provide the theoretical background for the microSIMPATY algorithm, the corresponding computational procedure, together with examples of successful application to several problem-specific contexts.