One of the main obstacles for proving Riemann-Roch for algebraic stacks is the lack of cohomology and homology theories that are closer to the K-theory and G-theory of algebraic stacks than the traditional cohomology and homology theories for algebraic stacks. In this paper we study in detail a family of cohomology and homology theories which we call Bredon-style theories that are of this type and in the spirit of the classical Bredon-cohomology and homology theories defined for the actions of compact topological groups on topological spaces. In a sequel to this paper, we establish Riemann-Roch theorems in this setting. We conclude with applications to virtual fundamental classes associated to dg-stacks: i.e. algebraic stacks (in general, of Artin type) provided with sheaves of commutative dgas, the main examples of which are moduli stacks of stable curves provided with a virtual structure sheaf associated to a perfect obstruction theory.