In the last few years, it has been discovered that topologically ordered phases of matter, such as fractional quantum Hall states and quantum spin liquids, can support a new class of topological defects, which are not intrinsic finite-energy quasiparticle excitations, but rather should be extrinsically imposed on the system by deforming the Hamiltonian. For systems in two spatial dimensions, these include line-like defects, which can occur at interfaces or boundaries of a topological phase, or point-like defects that occur at domain walls between different line-like defects. Not only have these discoveries led to new possible experimental proposals for creating topological qubits and probing electron fractionalization, but they have also been essential in the development of a new theoretical framework that can provide all possible topological invariants to characterize distinct topological phases of matter in the presence of arbitrary types of symmetries.