Electronic Journal of Differential Equations, Vol. 1999(1999), No. 25, pp. 1-14. Title: A necessary and sufficient condition for the diagonalization of multi-dimensional quasilinear systems Author: De-xing Kong (Kyoto Sangyo Univ., Kyoto, Japan) Abstract: In this paper, the author obtains a necessary and sufficient condition on the diagonalization of multi-dimensional quasilinear systems of first order, and gives some physical applications. Submitted July 1, 1999. Published July 29, 1999. Math Subject Classification: 35F20, 35L40. Key Words: Quasilinear systems; diagonalization; strict Riemann invariants; conservation laws.