Abstract. Palis conjectures that hyperbolic diffeomorphisms are C^1 dense in the complement C of the C^1 closure of the set of diffeomorphisms that has either a homoclinic tangency or a heterodimensional cycle. Towards the conjecture we prove that there is a C^1 residual subset R of diffeomorphisms such that every diffeomorphism in R is nearly hyperbolic. More precisely, we prove that every f in R has no simple minimally non-hyperbolic sets contained in the non-wandering set, and any non-simple minimally non-hyperbolic set of f must be partially hyperbolic of central direction at most 2-dimensional.