This talk will present a new method for computing the Igusa class polynomials of a primitive quartic $CM$ field. For a primitive quartic $CM$ field, $K$, we compute the Igusa class polynomials modulo $p$ for certain small primes $p$ and then use the Chinese remainder theorem and a conjectural bound on the denominators to construct the class polynomials. We also provide an extension to genus $2$ of the algorithm for determining endomorphism rings of elliptic curves. Our algorithm can be used to generate genus $2$ curves over a finite field with a given zeta function. \vskip .1in \noindent Joint work with Kirsten Eisentraeger, University of Michigan.