  We derive precise asymptotic estimates for the number of labelled graphs not
containing $K_{3,3}$ as a minor, and also for those which are edge maximal.
Additionally, we establish limit laws for parameters in random
$K_{3,3}$-minor-free graphs, like the expected number of edges. To establish
these results, we translate a decomposition for the corresponding graph class
into equations for generating functions and use singularity analysis. We also
find a precise estimate for the number of graphs not containing the graph
$K_{3,3}$ plus an edge as a minor.
