  The use of overconvergent cohomology in constructing $p$-adic $L$-functions,
initiated by Stevens and Pollack--Stevens in the setting of classical modular
forms, has now been established in a number of settings. The method is
compatible with constructions of eigenvarieties by Ash--Stevens, Urban and
Hansen, and is thus well-adapted to non-ordinary situations and variation in
$p$-adic families. In this note, we give an exposition of the ideas behind the
construction of $p$-adic $L$-functions via overconvergent cohomology.
Conditional on the non-abelian Leopoldt conjecture, we illustrate them by
constructing $p$-adic $L$-functions attached to families of base-change
automorphic representations for $\mathrm{GL}(2)$ over CM fields. As a
corollary, we prove a $p$-adic Artin formalism result for base-change $p$-adic
$L$-functions.
