Principal bundles and their associated fiber bundles famously play a foundational role in both algebraic and differential topology, as well as in fundamental and solid-state physics. More recently, their equivariant and higher homotopy enhancements (gerbes) have been crucial in generalized cohomology theory and for the physics of extended solitons and topological phases. This text is the first to offer a unified perspective of, and introduction to, these topics, providing an insight into material previously scattered across the literature. After a self-contained account of the classical theory of equivariant principal bundles in modern topological groupoid language, the book develops, on the novel backdrop of cohesive higher topos theory, a powerful theory of equivariant principal higher bundles. It establishes new methods like the 'smooth Oka principle' and 'twisted Elmendorf theorem' to elegantly prove classification results and clarify the relation to proper equivariant generalized cohomology theories.