This licentiate thesis consists of one paper about cohomology theories of algebraic varieties. Certain cohomology theories, such as singular cohomology of topological spaces, and sheaf cohomology, have a variant that we call “compactly supported cohomology”. Given an arbitrary cohomology theory, one can wonder what a compact support variant of this cohomology theory should be, and if it exists. In this paper we propose a definition of compactly supported cohomology theories on algebraic varieties. Using the language of ∞-toposes, we show that this variant exists for any cohomology theory of algebraic varieties that behaves well with abstract blowup squares. From this result we can derive classical results such as the existence of a weight filtration on compact support cohomology, and the existence of compactly supported homotopy algebraic K-theory.