Normal matrices are an important class of matrices; many authors discuss the normality of matrices. In terms of projection matrices, the inverse of matrices, one-sided $X$-equality, and $X$-idempotency of matrices, many new characterizations of normal matrices are obtained in this paper. It may be the first time that the projection, one-sided $X$-equality, and one-sided $X$-idempotency are used to characterize the normality of matrices.