The aim of this paper is to present some fixed point theorems for nowhere-normal-outward set-valued mappings. In Section I we first recall some definitions and auxiliary facts concerning properties of projection onto closed convex sets in Banach spaces. Next, we present a fixed point theorem for compact nowhere-normal-outward set-valued mappings. A more precise version of this result will be given in Section II for the case of Hilbert space. Finally, in Section III we give two fixed point theorems for non-compact set-valued mappings, which are condensing or non-expansive. The readers are referred to  [2], [3], [4], [5] for fixed point results, concerning set-valued mappings, satisfying other boundary conditions.