The paper deals with the existence and nonexistence of positive solutions of nonlinear elliptic equations, with the nonlinear term discontinuous in the unknown function. The authors distinguish between 3 types of solutions. Topological methods -- specifically the degree theory -- are employed to obtain the existence of solutions, either in \(\mathbb{R}^n\) or in a smooth bounded domain \(\Omega \subset \mathbb{R}^n\). When the nonlinearity is of sublinear form, the existence of a positive type I solution is proved. The existence of a positive type II solution is proved both for superlinear and sublinear (under an additional condition) nonlinearity. The nonexistence is proved in sublinear case with small enough range of discontinuity. The question of the existence of the solution of type I for general superlinear case is left open.