Ordinary materials are "passive" in the sense that their constituents are typically made by inert particles which are subjected to thermal fluctuations, internal interactions and external fields but do not move on their own. Living systems, like schools of fish, swarms of birds, pedestrians and swimming microbes are called "active matter" since they are composed of self-propelled constituents. Active matter is intrinsically in nonequilibrium and exhibits a plethora of novel phenomena as revealed by a recent combined effort of statistical theory, computer simulation and real-space experiments. After an introduction on biological and artificial microswimmers as key examples of active soft matter [1], the talk will focus on modelling of active Brownian particles and collective phenomena like motility-induced phase separation. Finally effects of inertia and the formation of active complexes will be discussed including coexistence of two states with different temperatures [2], an active refrigerator [3] and surfactants in active chiral fluids [4].

[1] For a review, see: C. Bechinger, R. di Leonardo,
H. Löwen, C. Reichhardt, G. Volpe, G. Volpe,
Active particles in complex and crowded environments,
Reviews of Modern Physics 88, 045006 (2016).

[2] S. Mandal, B. Liebchen, H. Löwen, Motility-induced temperature differences in coexisting phases,
Phys. Rev. Letters 123, 228001 (2019).

[3] L. Hecht, S. Mandal, H. Löwen, B. Liebchen, Active refrigerators powered by inertia,
Phys. Rev. Letters 129, 178001 (2022).

[4] C. Scholz, A. Ldov, T. Pöschel, M. Engel, H. Löwen,
Surfactants and rotelles in active chiral fluids, Science Advances 7, eabf8998 (2021).