In this series of seminars, I will give a gentle introduction to the differential geometric and Lie algebraic aspects of supergravity. I will first recall the description of supergravity theories in terms of Tanaka structures (non-holonomic G-structures) on supermanifolds and elucidate their relationship with a class of Z-graded Lie superalgebras g = g_{-2}+ g_{-1} + g_0 +g_1+g_2 of depth two. I will then present some recent results, in collaboration with J. Figueroa-O'Farrill (University of Edinburgh), on supergravity backgrounds and the algebraic structure of Lie superalgebras generated by Killing spinors. In the last part, I will recall the correspondence between different kinds of 3-algebras that recently appeared in Chern-Simons theories and Z-graded Lie superalgebras. The most general case is open and involves (generalized) Kantor triple systems and Lie superalgebras of depth two.