Abstract: Parabolic subalgebras and Z-gradings of complex Lie superalgebras are considered. For the classical (in the sense of V. Kac) simple complex Lie superalgebras, a complete description of parabolic subalgebras and Z-gradings in terms of the root systems is given. Gradings of these superalgebras by the root systems of their classical simple subalgebras are also discussed.