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We study the problem of efficiently constructing a curve C of genus 2 over afinite field F for which either the curve C itself or its Jacobian has aprescribed number N of F-rational points.  In the case of the Jacobian, we show that any `CM-construction' to producethe required genus-2 curves necessarily takes time exponential in the size ofits input.  On the other hand, we provide an algorithm for producing a genus-2 curve witha given number of points that, heuristically, takes polynomial time for mostinput values. We illustrate the practical applicability of this algorithm byconstructing a genus-2 curve having exactly 10^2014 + 9703 (prime) points, andtwo genus-2 curves each having exactly 10^2013 points.  In an appendix we provide a complete parametrization, over an arbitrary basefield k of characteristic neither 2 nor 3, of the family of genus-2 curves overk that have k-rational degree-3 maps to elliptic curves, including formulas forthe genus-2 curves, the associated elliptic curves, and the degree-3 maps.

Genus-2 curves and Jacobians with a given number of points