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A real square matrix $A$ of order $n \times n~ (n \geq 3)$ is called an$F_0$-matrix, if it is a $Z$-matrix (off-diagonal entries nonpositive), all ofwhose principal submatrices of orders at most $n-2$ are $M$-matrices whilethere is at least one principal submatrix of order $n-1$, which is an$N_0$-matrix. An $M$-matrix is a $Z$-matrix with the property that the realparts of all its eigenvalues are nonnegative. An $N_0$-matrix, in turn, ischaracterized by the fact that it is an invertible $Z$-matrix whose inverse is(entrywise) nonpositive. The first aim of this article is to present a shortsurvey of some subclasses of $Z$-matrices, pertinent to the second objective,where new results concerning $F_0$-matrices are presented.

A short survey of $Z$-matrices and new results on the subclass of $F_0$-matrices