We have presented the construction of a theory of distributions (generalized functions) with a "thick submanifold", that is, a new theory of thick distributions on ${\mathbb{R}}^n$ whose domain contains a smooth submanifold on which the test functions may be singular. We defined several operations, including "thick partial derivatives", and clarified their connection with their classical counterparts in Schwartz distribution theory. We also introduced and studied a number of special thick distributions, including new thick delta functions or, more generally, thick multilayer distributions along a submanifold.