The response to external mechanical forces is increasingly seen as a crucial aspect of cancer growth and a topic where the contribution of physics ideas and methods is important. Understanding if tumor progression towards increased malignancy reflects the geometry and mechanics of the microenvironment is an important issue still to be fully explored. In order to grow, tumors have to overcome the mechanical resistance posed by the tissues in which they originate, while cancer cells involved in metastasis are often subject to fluid pressure. Here we review the recent literature describing the role of solid and fluid pressure on tumor growth and progression. We discuss a variety of in vitro experiments as well as computational models used to interpret them. We conclude discussing future perspectives. © 2015, Società Italiana di Fisica and Springer-Verlag Berlin Heidelberg.