Let Lq(s. χ) be the Dirichlet L-function associated to χ, a cubic Dirichlet character with conductor of degree d over the polynomial ring Fq[T]. Following similar work by Keating and Snaith for moments of Riemann ζ-function, Conrey, Farmer, Keating, Rubinstein, and
Snaith [Con+05] introduced a framework for proposing conjectural formulae for integral moments of general L-functions with the help of random matrix theory.

In this thesis we review the heuristic found in [Con+05] and apply their work in order to propose moments for Lq(s, χ), cubic L-functions over function fields. We find asymptotic formulae when q ≡ 1 (mod 3), the Kummer case, and when q ≡ 2 (mod 3), the non-Kummer case. Moreover, while the authors of [Con+05] provide only the framework for proposing (k, k)-moments of primitive L-functions, we extend their work following the work of David, Lalin, and Nam to propose (k, l)-moments of cubic L-functions where k ≥ l ≥ 1 [DLN]. Furthermore, we provide explicit computations that elucidate the combinatorics of leading order moments and find a general form as well.