  It was shown by Brown and Henneaux that the classical theory of gravity on
AdS_3 has an infinite-dimensional symmetry group forming a Virasoro algebra.
More recently, Giveon, Kutasov and Seiberg (GKS) constructed the corresponding
Virasoro generators in the first-quantized string theory on AdS_3. In this
paper, we explore various aspects of string theory on AdS_3 and study the
relation between these two works. We show how semi-classical properties of the
string theory reproduce many features of the AdS/CFT duality. Furthermore, we
examine how the Virasoro symmetry of Brown and Henneaux is realized in string
theory, and show how it leads to the Virasoro Ward identities of the boundary
CFT. The Virasoro generators of GKS emerge naturally in this analysis. Our work
clarifies several aspects of the GKS construction: why the Brown-Henneaux
Virasoro algebra can be realized on the first-quantized Hilbert space, to what
extent the free-field approximation is valid, and why the Virasoro generators
act on the string worldsheet localized near the boundary of AdS_3. On the other
hand, we find that the way the central charge of the Virasoro algebra is
generated is different from the mechanism proposed by GKS.
