Abstract: Fractional quantum Hall systems are exotic quantum liquids which support quasiparticle excitations with fractional charge and even non-Abelian statistics. Theoretical investigations have found that these quasiparticles can be exploited to realize topological quantum computation that is fault-tolerant on the hardware level. In this article we describe the fractional quantum Hall effect with emphasis on its potential applications in topological quantum computation. We focus on the recent progress in the experimental detection of non-Abelian quasiparticles in the 5/2-filling quantum Hall system, as well as on some relevant theoretical interpretations.