The mechanical behavior of closed- and open-cell foams was analyzed using a computational homogenization strategy based on the finite element simulation of a representative volume element of the foam microstructure. The representation of the foam took into account the main microstructural features, namely density, fraction of solid material in cell walls and struts, cell anisotropy, cell size and distribution and strut shape. A parametric study was carried out to ascertain the influence of these parameters on the elastic modulus and the plateau stress under compression. It was found that these properties mainly depended on the density, fraction of solid material in cell walls and struts and cell anisotropy. Building upon the results of the parametric study, surrogate models were developed to relate the influence of the microstructure on the mechanical properties of the foams. It is believed that these models can be used for the design of foams with optimized properties for particular applications.