A matrix is said to have the Perron-Frobenius property if it has a positive dominant eigenvalue that corresponds to a nonnegative eigenvector. We study sets consisting of such matrices and give characterizations of these sets using their generalized eigenspaces. We also present some combinatorial, spectral, and topological properties, and the similarity transformations preserving the Perron-Frobenius property. In addition, certain results associated with nonnegative matrices are extended to matrices having the Perron-Frobenius property.