We study the quantum invariants of three-dimensional manifolds introduced by Reshetikhin and Turaev, following ideas laid out by Witten. In particular we consider the ungraded SU(2) and the even SO(3) version of the quantum theory.

This thesis consists of two parts. In the first we prove a conjecture of Larsen and Wang, that the SO(3) quantum invariants form a dense subset of the complex plane. The proof follows from properties of the quantum invariants and from a theorem of Larsen and Wang which states that the corresponding SO(3) quantum representation has dense image.

In the second part we investigate a theorem of Garoufalidis, which shows that both the SU(2) and SO(3) quantum invariants provide lower bounds on the Heegaard genus. We compare the various lower bounds from the quantum invariants with the first betti number (both with [special characters omitted] and [special characters omitted]/2[special characters omitted] coefficients) and the rank of the fundamental group. We simplify a proof of Boileau and Zieschang, and in so doing we further show that the quantum bound on Heegaard genus can be strictly larger than the lower bound provided by the rank of the fundamental group.