A real square matrix A of order \(n \times n~ (n \ge 3)\) is called an \(F_0\) -matrix, if it is a Z-matrix (off-diagonal entries nonpositive), all of whose principal submatrices of orders at most \(n-2\) are M-matrices, while there 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 real parts of all its eigenvalues are nonnegative. An \(N_0\) -matrix, in turn, is characterized 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 survey of some subclasses of Z-matrices, pertinent to the second objective, where new results concerning \(F_0\) -matrices are presented.