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Abstract: How many different ways can we arrange n convex sets in R^d? One answer is provided by counting the number of d-representable complexes on vertex set [n]. We show that there are exp(Theta(n^d log n)) many such complexes, and provide bounds on the constants involved. As a consequence, we show that d-representable complexes comprise a vanishingly small fraction of d-collapsible complexes. In the case d=1 our results are more precise, and improve the previous best estimate for the number of interval graphs. These results are joint work with Boris Bukh.