Biomaterials for tissue scaffolds are key components in modern tissue engineering and regenerative medicine. Targeted reconstructive therapies require a proper choice of biomaterial and an adequate choice of cells to be seeded on it. The introduction of stem cells, and the transdifferentiation procedures, into regenerative medicine opened a new era and created new challenges for modern biomaterials. They must not only fulfill the mechanical functions of a scaffold for implanted cells and represent the expected mechanical strength of the artificial tissue, but furthermore, they should also assure their survival and, if possible, affect their desired way of differentiation. This paper aims to review how modern biomaterials, including synthetic (i.e., polylactic acid, polyurethane, polyvinyl alcohol, polyethylene terephthalate, ceramics) and natural (i.e., silk fibroin, decellularized scaffolds), both non-biodegradable and biodegradable, could influence (tissue) stem cells fate, regulate and direct their differentiation into desired target somatic cells.