  A major current goal of noncommutative geometry is the classification of
noncommutative projective surfaces. The generic case is to understand algebras
birational to the Sklyanin algebra. In this thesis we complete a considerable
component of this problem. Let $S$ denote the 3-dimensional Sklyanin algebra
over an algebraically closed field, and assume that $S$ is not a finite module
over its centre. In earlier work Rogalski, Sierra and Stafford classified the
maximal orders inside the 3-Veronese $S^{(3)}$ of $S$. We complete and extend
their work and classify all maximal orders inside $S$. As in Rogalski, Sierra
and Stafford's work, these can be viewed as blowups at (possibly non-effective)
divisors. A consequence of this classification is that maximal orders are
automatically noetherian among other desirable properties. This work both
relies upon, and lends back to, the work of Rogalski, Sierra and Stafford. In
particular, we also provide a converse for their classification of maximal
orders in the higher Veronese subrings $S^{(3n)}$ of $S^{(3)}$.
