The problem of identifying weak sharp minima of order m, an important class of (possibly) nonisolated minima, is investigated in this paper. Some sufficient conditions for weak sharp minimality in nonsmooth mathematical programming are presented, and some characterizations of weak sharp minimality are obtained, with special attention given to orders one and two. It is also demonstrated that two of these sufficient conditions guarantee exactness of an l(1) penalty function. A key role in this paper is played by two geometric concepts: the limiting proximal normal cone and a generalization of the contingent cone.