Due to recent experimental efforts on light muonic atoms, we consider the quantum electrodynamics (QED) theory of two-body muonic atoms, namely, of those with $A=3$. Here we present the results for the Lamb shift and fine structure up to the order ${\ensuremath{\alpha}}^{5}m$ for the $n=2$ levels, which in particular include (1) pure QED contributions, (2) the coefficient to the ${r}_{N}^{2}$ term with QED corrections (where ${r}_{N}$ is the rms nuclear charge radius), and (3) the general expressions for the nuclear-structure contributions consistent with the presented QED theory. We revisit theory for the muonic helium-3 ion by rechecking all the relevant theoretical contributions and develop a theory of muonic tritium. We also reestimate the uncertainty of the nuclear-structure contribution.