Fractional descriptor systems belong to an important class of fractional differential systems. The systems have a wide range of applications in physical and engineering problems such as integrated circuits with superconductor materials. In this paper, a new class of fractional descriptor systems is considered and solvability of the fractional descriptor systems is discussed. Furthermore, explicit solutions of the fractional descriptor systems are given and sufficient conditions for the positivity of the fractional descriptor systems are established. An example is presented to illustrate the feasibility and applications of the obtained results. Our results can be extended to the case of fractional descriptor systems with different fractional orders.