I will discuss the revolutionary new concept of topological quantum computation, which is fault-tolerant at the hardware level with no need, in principle, of any quantum error correction protocols. Errors simply do not happen since the physical qubits and the computation steps are protected against decoherence by non-local topological correlations in the underlying physical system. The key idea is non-Abelian statistics of the quasiparticles (called 'anyons' as opposed to fermions or bosons), where the space-time braiding of the anyons around each other, i.e. quantum 'knots', form topologically protected quantum gate operations. I will describe in detail the status of the subject by discussing the theoretical principles guiding the experimental search for the appropriate topological phases of matter where such non-Abelian anyons may exist. Among the most significant possibilities are certain even-denominator fractional quantum Hall states, exotic chiral p-wave superconductors, sandwich structures made from superconductors/ semiconductors or superconductors/insulators, and suitably designed cold atomic systems. In this context, I will also discuss the race to find the mysterious Majorana fermions in solid state systems, with the Majorana fermions being the simplest generic examples of non-Abelian objects in nature. I will explain how the subject of topological quantum computation synergistically brings together conformal field theory and advanced mathematics on one hand with materials science and quantum information on the other.