The inverse Galois problem asks whether every finite group appears as the Galois group of some finite extension of $\mathbb Q$. I was wondering to what extent the analogous problem for ideal class groups has been investigated. More precisely, consider the following question:

Is every finite abelian group the ideal class group of some number field (finite extension of $\mathbb Q$)?

I'd be interested to hear about any partial results, as I suppose this question is still open. I'd be also interested in any results about a weaker problem:

Is every positive integer the ideal class number of some number field?

Again, any reference, even to a partial result, will be appreciated.