Differential graded (dg) algebra structures on complexes, and more particularly on free resolutions, have played a key role in commutative algebra. Some of the history will be mentioned, from its origins in algebraic topology to the motivations for having such structures on minimal free resolutions. In addition, we will discuss some general constructions producing (mostly non-minimal) resolutions with dg algebra structures. Time permitting, we may very briefly mention the related roles of A-infinity structures and Massey operations.