Summary: The paper proposes some new concepts of higher order feasible (descent) directions, higher order quasifeasible (quasidescent) directions and higher order tangent directions. Using the support functions of these sets of higher order directions, higher orer necessary conditions for a nonsmooth optimization problem (P) in a topological vector space are established. Equivalent representations of the above-mentioned sets and relations between these sets and higher order variational sets as described by K. H. Hoffmann and H.-J. Kornstaedt [J. Optimization Theory Appl. 26, 533-568 (1978; Zbl 0373.90066), A. Linnemann [ibid. 38, 483-511 (1982; Zbl 0471.49019)] and H.-J. Kornstaedt [in: Semi-infinite programming, Proc. Workshop Bad Honnef 1978, Lect. Notes Control Inf. Sci. 15, 31-50 (1979; Zbl 0418.49023)] are discussed under Fréchet differentiability assumptions. As a result higher order necessary conditions for smooth optimization problems are obtained which extend many of the known higher and lower order necessary optimality conditions for smooth problems.