Let A[ω]ω be a maximal almost disjoint family and assume P is a forcing notion. Say A is P-indestructible if A is still maximal in any P-generic extension. We investigate P-indestructibility for several classical forcing notions P. In particular, we provide a combinatorial characterization of P-indestructibility and, assuming a fragment of MA, we construct maximal almost disjoint families which are P-indestructible yet Q-destructible for several pairs of forcing notions . We close with a detailed investigation of iterated Sacks indestructibility.

We present a survey of some results and problems concerning constructions which require a diagonalization of length continuum to be carried out, particularly constructions of almost disjoint families of various sorts. We emphasize the role of cardinal invariants of the continuum and their combinatorial characterizations in such constructions.