Bauer's Theorem (a simple consequence of the Chebotarev Density Theorem) states that a finite
Galois extension K of an algebraic number field F is uniquely determined (as a subield of
some fixed algebraic closure of F) by the set of primes of F which split completely in K.  Thus knowing all possible Galois groups is the same as knowing all possible splitting laws
in finite Galois extensions.  Being able to describe these splitting laws in some explicit
fashion is basically "nonabelian reciprocity", which is THE most important problem in algebraic number theory, so the "inverse Galois problem" is of FUNDAMENTAL importance to all
number theorists.