The authors consider generalized vector distributions as locally finitely generated \(C^\infty_c(M)\) submodules \(D\subset\Gamma(TM)\). Evaluation gives usual family of subspaces of the tangent bundle, whose dimensions need not be constant (but are semi-continuous). The first theorem states that any such smooth distribution \((M,\mathcal{D})\) possesses a Riemannian structure, which generalizes a sub-Riemannian structure on regular vector distributions. Then using the standard idea of sums of squares and certain equivalence relations, the authors introduce the horizontal Laplacian \(\Delta_{\mathcal{D}}\) and prove that for compact \(M\) it is essentially self-adjoint and longitudinally hypoelliptic (within the smallest singular foliation induced by the distribution). This generalizes the Hörmander’s result on the hypoellipticity in sub-Riemannian geometry.