  This is a survey paper on cohomology theories for $A_\infty$ and $E_\infty$
ring spectra. Different constructions and main properties of topological
Andr\'e-Quillen cohomology and of topological derivations are described. We
give sample calculations of these cohomology theories and outline applications
to the existence of $A_\infty$ and $E_\infty$ structures on various spectra. We
also explain the relationship between topological derivations, spaces of
multiplicative maps and moduli spaces of multiplicative structures.
