This paper deals with the construction of conservative high order and positivity preserving schemes for nonlinear hyperbolic conservation laws. In particular, we consider space-time Petrov-Galerkin discretizations inspired by residual distribution ideas and based on a PkPm polynomial approximations in space-time. The approximation is continuous in space and discontinuous in time so that one single space-time slab at the time can be dealt with. We show constructions involving linear high order and nonlinear schemes. Principles borrowed from the residual distribution approach, such as multidimensional upwinding and positivity preservation, are used to construct the Petrov-Galerkin test functions. The numerical results on one dimensional linear and nonlinear conservation laws show that higher accuracy and positivity are obtained uniformly with respect to the physical CFL number
