I will present ongoing work with J. Figueroa-O’Farrill and P. de Medeiros on the algebraic structure of Lie superalgebras $g = g_0 \oplus g_1$ generated by Killing spinors. I will explain how any $g$ can be regarded as an appropriate deformation of a subalgebra of the Poincaré superalgebra and discuss applications to the classification of supersymmetric supergravity backgrounds and the geometries admitting rigidly supersymmetric field theories.