Necessary and sufficient conditions are given in order that a matrix A have a nonnegative generalized inverse $A^ + $. The concept of row-monotonicity is introduced and a characterization of row-monotone matrices is used to derive a necessary and sufficient condition on $A\geqq 0$ so that $A^ + \geqq 0$. Applications of these results include the special case where A has full column rank, thus yielding results of Collatz, Mangasarian and Ben-Israel as corollaries.