  A group is $\textit{finitely axiomatizable}$ (FA) in a class $\mathcal{C}$ if
it can be determined up to isomorphism within $\mathcal{C}$ by a sentence in
the first-order language of group theory. We show that profinite groups of
various kinds are FA in the class of profinite groups. Reasons why certain
groups cannot be FA are also discussed.
