The quantum supergroup OSPq(1\2n) is studied systematically. A Haar functional is constructed, and an algebraic version of the Peter-Weyl theory is extended to this quantum supergroup. Quantum homogeneous superspaces and quantum homogeneous supervector bundles are defined following the strategy of Connes' theory. Parabolic induction is developed by employing the quantum homogeneous supervector bundles. Quantum Frobenius reciprocity and a generalized Borel-Weil theorem are established for the induced representations. (C) 1999 American Institute of Physics. [S0022-2488(99)00205-4].