Active matter agents consume internal energy or extract energy from the environment for locomotion and force generation. Already, rather generic models, such as ensembles of active Brownian particles, exhibit phenomena, which are absent at equilibrium, particularly motility-induced phase separation and collective motion. Further intriguing nonequilibrium effects emerge in assemblies of bound active agents as in linear polymers or filaments. The interplay of activity and conformational degrees of freedom gives rise to novel structural and dynamical features of individual polymers, as well as in interacting ensembles. Such out-of-equilibrium polymers are an integral part of living matter, ranging from biological cells with filaments propelled by motor proteins in the cytoskeleton and RNA/DNA in the transcription process to long swarming bacteria and worms such as Proteus mirabilis and Caenorhabditis elegans, respectively. Even artificial active polymers have been synthesized. The emergent properties of active polymers or filaments depend on the coupling of the active process to their conformational degrees of freedom, aspects that are addressed in this article. The theoretical models for tangentially and isotropically self-propelled or active-bath-driven polymers are presented, both in the presence and absence of hydrodynamic interactions. The consequences for their conformational and dynamical properties are examined, with emphasis on the strong influence of the coupling between activity and hydrodynamic interactions. Particular features of emerging phenomena in semi-dilute systems, induced by steric and hydrodynamic interactions, are highlighted. Various important, yet theoretically unexplored, aspects are featured, and future challenges are discussed.