2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/192176We 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.)36 pages, LaTeX, RIMS-990, YITP/K-1087, YITP/U-94-25, SULDP-1994-7High Energy Physics - Theorytext