A continuous action of a topological group $G$ on a topological space $X$ has a section if there is a continuous map $s: X/G \to X$ such that $p s=id_X$ (and $p$ is the quotient map $p: X\to X/G$).

The class of such actions contain both simple complexes of groups (described for example in Bridson-Haefliger book), quasitoric spaces and many more examples.

I will discuss constructions of such actions in various contexts, and show that one can compute for them interesting invariants, such as equivariant cohomology, or (sometimes) cohomology.