The previous answers are all on point; let me just say a little more.

First, the inverse Galois problem (IGP) as a problem is a *sink*, not a *source* (or something with both inward and outward flow!): I know of no nontrivial consequences of assuming that every finite group over ${\bf Q}$ (or even over every Hilbertian field) is a Galois group. This does not mean it's a bad problem: the same holds for Fermat's Last Theorem (FLT).

As with FLT, if IGP were *easy* to prove, then it would be of little interest. As a good example, if you know Dirichlet's theorem on primes in arithmetic progressions, it's easy to prove that every finite abelian group occurs as a Galois group of ${\bf Q}$. What does this tell you about the maximal abelian extension of ${\bf Q}$? Not much. The Kronecker-Weber theorem is an order of magnitude deeper. But as with FLT, the special cases of IGP that have been established use a wide array of fascinating techniques and provide an important border-crossing between algebra and geometry.

Arguably more interesting than IGP itself is the Regular Inverse Galois Problem (RIGP):

For *any* field $K$ and any finite group $G$, there exists a regular function field $K(C)/K(t)$ with Galois group isomorphic to $G$.

If $K$ is Hilbertian (e.g. a global field) then RIGP for $K$ implies IGP for $K$. Now RIGP is of great interest in arithmetic geometry: given any finite group $G$ there are infinitely many moduli spaces (Hurwitz spaces) attached to the problem of realizing $G$ regularly over $K$ (because we have discrete invariants which can take infinitely many possible values, like the number of branch points). If even one of these Hurwitz schemes has a $K$-rational point, then $G$ occurs regularly over $K$. In general, the prevailing wisdom about varieties over fields like ${\bf Q}$ is that they should have very few rational points other than the ones that stare you in the face. (Yes, it is difficult or impossible to formalize this precisely.) So it is somewhat reasonable to say that the chance that a given Hurwitz space, say of general type, has a ${\bf Q}$-rational point is zero, but what about the chance that *at least one* of infinitely many Hurwitz spaces, related to each other by various functorialities, has a ${\bf Q}$-rational point? To me that is one of mathematics' most fascinating questions: to learn the answer *either way* would be tremendously exciting.

stilldon't know which groups arise as the Galois group of an irreducible polynomial over Q; and someone asks why should we care??? $\endgroup$ – JS Milne Jan 5 '10 at 5:17