A complete classification of evolution equations u t =F(x, t, u, u x , ..., u x k ) which describe pseudo-spherical surfaces, is given, thus providing a systematic procedure to determine a one-parameter family of linear problems for which the given equation is the integrability condition. It is shown that for every second-order equation which admits a formal symmetry of infinite rank ( formal integrability) such a family exists (kinematic integrability). It is also shown that this result cannot be extended as proven to third-order formally integrable equations. This fact notwithstanding, a special case is proven, and moreover, several equations of interest, including the Harry Dym, cylindrical KdV, and a family of equations solved by inverse scattering by Calogero and Degasperis, are shown to be kinematically integrable. Conservation laws of equations describing pseudo-spherical surfaces are studied, and several examples are given. 1998 Academic Press This structure was considered for the first time by Chern and Tenenblat [4], motivated by Sasaki's [15] observation that the equations which are the necessary and sufficient condition for the integrability of a linear problem of AKNS type (Ablowitz et al. [2]) do describe pseudo-spherical surfaces. Its importance, in the present context, arises from the fact that article no. DE983430 195 0022-0396Γ98 25.00