  This thesis is a study of two dimensional noncritical string theory. The main
tool which is used, is the matrix model. Introductions to both the Liouville
model and its matrix model formulation are included. In particular the special
states are discussed. Some calculations of partition functions on genus one
using field theory techniques are given. Nonperturbative issues and string
theory at finite radius are discussed. Zero momentum correlation functions are
calculated using the matrix model. One important result is a set of recursion
relations. The treatment is extended to nonzero momentum. The main result is a
clear identification of the special states. Some comments on the Wheeler de
Witt equation is given. The matrix model $W_{\infty}$ algebra is introduced.
This organizes the previous results. In particular, a simple derivation of the
genus zero tachyon correlation functions is given. The results are then
extended to higher genus. It is seen how a deformation of the algebra is
responsible for much of the higher genus structure. Some very explicit formulae
are derived. Then the Liouville and matrix model calculations are compared
followed by some general conclusions.
