The geometry of (super) conformal quantum mechanics

N-particle quantum mechanics described by a sigma model with an N-dimension al target space with torsion is considered. It is shown that an SL(2, R) co nformal symmetry exists if and only if the geometry admits a homothetic Kil ling vector D(a)partial derivative(a) whose associated one-form D(a)dX(a) i s closed. Further, the SL(2, R) can always be extended to Osp(1/2) supercon formal symmetry, with a suitable choice of torsion, by the addition of N re al fermions. Extension to SU(1, 1/1) requires a complex structure I and a h olomorphic U(1) isometry D(a)I(a)(b)partial derivative(b). Conditions for e xtension to the superconformal group D(2, 1; alpha), which involve a triple t of complex structures and SU(2) x SU(2) isometries, are derived. Examples are given.