We show that an ordinary semiconducting thin film with spin-orbit coupling can, under appropriate circumstances, be in a quantum topologically ordered state supporting exotic Majorana excitations which follow non-Abelian statistics. The key to the quantum topological order is the coexistence of spin-orbit coupling with proximity-induced s-wave superconductivity and an externally induced Zeeman coupling of the spins. For the Zeeman coupling below a critical value, the system is a nontopological (proximity-induced) s-wave superconductor. However, for a range of Zeeman coupling above the critical value, the lowest energy excited state inside a vortex is a zero-energy Majorana fermion state. The system, thus, has entered into a non-Abelian s-wave superconducting state via a topological quantum phase transition (TQPT) tuned by the Zeeman coupling. In the topological phase, since the time-reversal symmetry is explicitly broken by the Zeeman term in the Hamiltonian, the edge of the film constitutes a chiral Majorana wire. Just like the s-wave superconductivity, the Zeeman coupling can also be proximity induced in the film by an adjacent magnetic insulator. We show this by an explicit model tight-binding calculation for both types of proximity effects in the heterostructure geometry. Here we show that the same TQPT can be accessed by varying the interface transparency between the film and the superconductor. For the transparency below (above) a critical value, the system is a topological (regular) s-wave superconductor. In the one-dimensional version of the same structure and for the Zeeman coupling above the critical value, there are localized Majorana zero-energy modes at the two ends of a semiconducting quantum nanowire. In this case, the Zeeman coupling can be induced more easily by an external magnetic field parallel to the wire, obviating the need for a magnetic insulator. We show that, despite the fact that the superconducting pair potential in the nanowire is explicitly s wave, tunneling of electrons to the ends of the wire reveals a pronounced zero-bias peak. Such a peak is absent when the Zeeman coupling is below its critical value, i.e., the nanowire is in the nontopological s-wave superconducting state. We argue that the observation of this zero-bias tunneling peak in the semiconductor nanowire is possibly the simplest and clearest experiment proposed so far to unambiguously detect a Majorana fermion mode in a condensed-matter system.