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The recent paper by Homlin, Jones, Kurlberg, McLeman, and Petersen (Experimental Math., to appear) is devoted to these questions especially in the context of imaginary quadratic fields. One should expect that every natural number arises as the class number of an imaginary quadratic field. Refining an earlier conjecture of Soundararajan, in this paper a precise asymptotic is formulated for the number of imaginary quadratic fields with a given class number. They also formulate conjectures on what kind of groups can arise as class groups of imaginary quadratic fields -- for example, one expects that for any odd prime $p$, $({\Bbb Z}/p{\Bbb Z})^3$ is not the class group of any imaginary quadratic field. The paper gives much data on such questions together with many related references.