Conducting fibres and yarns promise to become an essential part of the next generation of wearable electronics that seamlessly integrate electronic function into one of the most versatile and most widely used form of materials: textiles. This review explores the many types of conducting fibres and yarns that can be realised with conjugated polymers and carbon materials, including carbon black, carbon nanotubes and graphene. We discuss how the interplay of materials properties and the chosen processing technique lead to fibres with a wide range of electrical and mechanical properties. Depending on the choice of conjugated polymer, carbon nanotube, graphene, polymer blend, or nanocomposite the electrical conductivity can vary from less than 10⁻³ to more than 10³ S cm⁻¹, accompanied by an increase in Young's modulus from 10 s of MPa to 100 s of GPa. Further, we discuss how conducting fibres can be integrated into electronic textiles (e-textiles) through e.g. weaving and knitting. Then, we provide an overview of some of the envisaged functionalities, such as sensing, data processing and storage, as well as energy harvesting e.g. by using the piezoelectric, thermoelectric, triboelectric or photovoltaic effect. Finally, we critically discuss sustainability aspects such as the supply of materials, their toxicity, the embodied energy of fibre and textile production and recyclability, which currently are not adequately considered but must be taken into account to ready carbon based conducting fibres for truly practical e-textile applications.