Quasi-invariants are natural geometric generalizations of invariant polynomials of finite reflection groups. They first appeared in mathematical physics in the early 1990s, and since then have found applications in many other areas: most notably, representation theory, algebraic geometry and combinatorics. In this talk, I will explain how the algebras of quasi-invariants can be realized topologically as (equivariant) cohomology rings of certain spaces naturally attached to compact connected Lie groups. Our main result is a generalization of a well-known theorem of A. Borel that realizes the algebra of invariant polynomials of a Weyl group W as the cohomology ring of the classifying space BG of the corresponding Lie group G. Replacing equivariant cohomology with equivariant K-theory (resp., elliptic cohomology) gives multiplicative (resp., elliptic) analogues of quasi-invariants. But perhaps most interesting is the fact that the spaces of quasi-invariants can be also defined for non-Coxeter (p-adic) pseudo-reflection groups, in which case the compact Lie groups are replaced by p-compact groups – remarkable homotopy-theoretic objects a.k.a. homotopy Lie groups. Time permitting, I will also discuss some applications in the context of stable homotopy theory. (Based on joint work with A. C. Ramadoss).