  We review the recent development in the representation theory of the
$W_{1+\infty}$ algebra. The topics that we concern are, Quasifinite
representation, Free field realizations, (Super) Matrix Generalization,
Structure of subalgebras such as $W_\infty$ algebra, Determinant formula,
Character formula. (Invited talk at ``Quantum Field Theory, Integrable Models
and Beyond", YITP, 14-17 February 1994. To appear in Progress of Theoretical
Physics Proceedings Supplement.)
