In this talk I will discuss $\mathcal{D}$-stability for strongly self absorbing C*-algebras $\mathcal{D}$ and its relevance for the classification of C*-algebras. I will then discuss an equivariant notion of $\mathcal{D}$-stability with respect to an action of a tensor category and an adaptation of a classical result of Kirchberg to this setting. This talk will not assume any knowledge on tensor categories and their actions.