Cell growth describes an essential feature of biological tissues. This growth process may be modeled by using a set of relatively simple governing equations based on the axioms of mass and momentum balance, and using a continuum framework that describes cells and tissues as mixtures of a solid matrix, a solvent and multiple solutes. In this model the mechanics of cell growth is driven by osmotic effects, regulated by the cells' active uptake of solutes and passive uptake of solvent. By accounting for the anisotropy of the cells' cytoskeletal structures or extracellular matrix, as well as external constraints, a wide variety of growing shapes may be produced as illustrated in various examples.