Diffeology, introduced around 1980 by Jean-Marie Souriau following earlier work of Kuo-Tsai Chen, gives a simple way to extend notions of differential topology beyond manifolds. A diffeology on a set specifies which maps from open subsets of Euclidean spaces to the set are "smooth". Examples include (possibly non-Hausdorff) quotients of manifolds and (infinite dimensional) spaces of smooth mappings between manifolds. The symplectic reduction procedure yields particularly interesting spaces, which come with a stratified symplectic structure; there are intriguing open questions about the relation of their intrinsic diffeology with their structure as stratified spaces. I will present a sample of examples, results, and questions, to give you a taste of the subject.