The number of imaginary quadratic fields with a given class number

We investigate the number ${\Cal F}(h)$ of imaginary quadratic fields with class number $h$. We establish an asymptotic formula for the average value of ${\Cal F}(h)$. We also establish a modest non-trivial upper bound for ${\Cal F}(h)$ and give an application to a question of Rosen and Silverman on the odd part of the class number. Finally, we speculate on the asymptotic nature of ${\Cal F}(h)$.