  In view of one day proving the AdS/CFT correspondence, a deeper understanding
of string theory on certain curved backgrounds such as AdS_5xS^5 is required.
In this dissertation we make a step in this direction by focusing on RxS^3. It
was discovered in recent years that string theory on AdS_5xS^5 admits a Lax
formulation. However, the complete statement of integrability requires not only
the existence of a Lax formulation, but also that the resulting integrals of
motion are in pairwise involution. This idea is central to the first part of
this thesis. Exploiting this integrability we apply algebro-geometric methods
to string theory on RxS^3 and obtain the general finite-gap solution. The
construction is based on an invariant algebraic curve previously found in the
AdS_5xS^5 case. However, encoding the dynamics of the solution requires
specification of additional marked points. By restricting the symplectic
structure of the string to this algebro-geometric data we derive the
action-angle variables of the system. We then perform a first-principle
semiclassical quantisation of string theory on RxS^3 as a toy model for strings
on AdS_5xS^5. The result is exactly what one expects from the dual gauge theory
perspective, namely the underlying algebraic curve discretises in a natural
way. We also derive a general formula for the fluctuation energies around the
generic finite-gap solution. The ideas used can be generalised to AdS_5xS^5.
