We investigate the question "What C*-algebras embed into the Calkin algebra Q(H)?". We present a result from a joint work with Ilijas Farah and Georgios Katsimpas which entails that assuming Martin's axiom, a strengthening of the Baire Category Theorem known to be independent from the standard axiomatization of set theory ZFC, then all C*-algebras of density character strictly smaller than continuum embed into Q(H). We move then to C*-algebras of density continuum, and we show that both the reduced C*-algebra generated by the free group on continuum generators, and the tensor product of continuum copies of the C*-algebra of two by two matrices embed into Q(H).