Let M be an n-dimensional compact connected Riemannian manifold with non- empty boundary. We say that the boundary is p-convex (where p is an integer with \(1\leq p\leq n-1)\) if at each point the sum of any p principal curvatures, defined with respect to the inward normal, is positive. This condition together with non-negative sectional curvature in the interior has strong implication for the topology of the manifold. The main result of this paper is the following. If M carries a Riemannian metric with non-negative sectional curvature and p-convex boundary, then M has the homotopy type of a CW-complex of dimension \(\leq (p-1)\). A converse of this theorem is proved in a forthcoming paper by the same author.