A strongly self-absorbing C ∗ -algebra is a unital, separable infinite dimensional C ∗ -algebra D satisfying that the first coordinate embedding from D into D ⊗ D is approximately unitarily equivalent to an isomorphism. A C ∗ -algebra A is said to be D-absorbing if A ∼= A ⊗ D. In this talk, I will survey some results concerning permanence of D-absorption under formation of crossed products and continuous fields. This is joint work with M. Rordam and W. Winter.