In the present work a general theoretical framework for coupled dimensionally-heterogeneous partial differential equations is developed. This is done by recasting the variational formulation in terms of coupling interface variables. In such a general setting we analyze existence and uniqueness of solutions for both the continuous problem and its finite dimensional approximation. This approach also allows the development of different iterative substructuring solutionmethodologies involving dimensionally-homogeneous subproblems. Numerical experiments are carried out to test our theoretical results.