An analytic theory is presented for the foundering of the lithosphere at the onset of subduction. The two-dimensional cartesian model calculates the instantaneous response of a viscous half-space to a surface load that corresponds to the negative buoyancy of a semi-infinite plate. In the early stages of foundering, just after the mantle floods onto the top of the lithosphere, the edge of the slab sinks into the mantle with a downward velocity uz given by (1/6e)(Δ&rgr;/&rgr;)4(&rgr;gd2/&mgr;)(d/xn)2, where &rgr; is the density of the mantle, Δ&rgr; is the density of the lithosphere minus the density of the mantle, d is the thickness of the lithosphere, g is the acceleration of gravity, &mgr; is the viscosity of the mantle, and xn is the flexural length scale of the lithosphere. Alternatively, uz is given by (3/e)(Δ&rgr;/&rgr;)3(d/xn)2uz (Stokes), where uz (Stokes) is the Stokes velocity of fall of a sphere of radius d/2 with excess density Δ&rgr; through a fluid of viscosity &mgr; and density &rgr;. The initial subsidence velocity of the edge of a lithospheric slab is much slower than the sinking velocity of a similar-sized and negatively buoyant sphere. For typical values of the above parameters, the edge of the plate sinks into a mantle of viscosity 1018 Pa s at a rate of about 7.5 mm yr-1. Lithospheric foundering is facilitated by low mantle viscosity and old, thick plates.¿ 1997 American Geophysical Union