The study of the Schrödinger equation with the position-dependent effective mass has attracted a lot of attention, due to its applications in many fields of physics, including the properties of the semiconductors, semiconductor heterostruc-tures, graded alloys, quantum liquids, Helium-3 clusters, quantum wells, wires and dots etc. In the present work we obtain several classes of solutions of the one-dimensional Schrödinger equation with position-dependent particle mass. As a first step the single particle Schrödinger equation with position-dependent mass is transformed into an equivalent Riccati type equation. By considering some integrability cases of the Ric-cati equation, seven classes of exact analytical solutions of the Schrödinger equation are obtained, with the particle mass function and the external potential satisfying some consistency conditions.