We examine the structure of spacetime symmetries of toroidally compactified
string theory within the framework of noncommutative geometry. Following a
proposal of Frohlich and Gawedzki, we describe the noncommutative string
spacetime using a detailed algebraic construction of the vertex operator
algebra. We show that the spacetime duality and discrete worldsheet symmetries
of the string theory are a consequence of the existence of two independent
Dirac operators, arising from the chiral structure of the conformal field
theory. We demonstrate that these Dirac operators are also responsible for the
emergence of ordinary classical spacetime as a low-energy limit of the string
spacetime, and from this we establish a relationship between T-duality and
changes of spin structure of the target space manifold. We study the
automorphism group of the vertex operator algebra and show that spacetime
duality is naturally a gauge symmetry in this formalism. We show that classical
general covariance also becomes a gauge symmetry of the string spacetime. We
explore some larger symmetries of the algebra in the context of a universal
gauge group for string theory, and connect these symmetry groups with some of
the algebraic structures which arise in the mathematical theory of vertex
operator algebras, such as the Monster group. We also briefly describe how the
classical topology of spacetime is modified by the string theory, and calculate
the cohomology groups of the noncommutative spacetime. A self-contained,
pedagogical introduction to the techniques of noncommmutative geometry is also
included.