Abstract The two function theories of monogenic and of slice monogenic functions with values in a Clifford algebra have been extensively studied in the literature. Although these functions are very different in nature they can be related via the celebrated Fueter-Sce mapping theorem or via Radon transform. In this talk we introduce a new class of functions, that of generalized partial-slice monogenic functions, which includes both of them as special cases. For this class of functions we can prove main properties such as the identity theorem, a Representation Formula, two different types of extension theorems, the Cauchy integral formula. The talk is based on a joint work with Zhenghua Xu.