Three-dimensional lattices have applications across a range of fields including structural lightweighting, impact absorption and biomedicine. In this work, lattices based on triply periodic minimal surfaces were produced by polymer additive manufacturing and examined with a combination of experimental and computational methods. This investigation elucidates their deformation mechanisms and provides numerical parameters crucial in establishing relationships between their geometries and mechanical performance. Three types of lattice were examined, with one, known as the primitive lattice, being found to have a relative elastic modulus over twice as large as those of the other two. The deformation process of the primitive lattice was also considerably different from those of the other two, exhibiting strut stretching and buckling, while the gyroid and diamond lattices deformed in a bending dominated manner. Finite element predictions of the stress distributions in the lattices under compressive loading agreed with experimental observations. These results can be used to create better informed lattice designs for a range of mechanical and biomedical applications