Austrian Science Fund (FWF), Lise-Meitner Fellowship No. M 2470-N28 (Jan Smrek);Italian MIUR grant Rita Levi-Montalcini (Emanuele Locatelli)Polish National Science Centre under grant OPUS 19, grant no. 2020/37/B/ST3/01471 (Franco Ferrari);European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 947918, TAP) and Royal Society University Research Fellowship (Davide Michieletto);EMBO Installation Grant 2057, the Polish Ministry for Science and Higher Education0003/ID3/2016/64, Polish National Science Centre: COVID-19 EXPRESS CALL and OPUS 16, 2018/31/B/NZ1/04016 (Joanna I. Su\u0142kowska);TEAM programme of the Foundation for Polish Science co-financed by the European Union under the European Regional Development Fund POIR.04.04.00-00-5C55/17-00 (Piotr Su\u0142kowski);Grant KP-06-COST-9/2019 of the Bulgarian National Science Fund, a National Co-funding for participation in approved COST Actions (Nevena Ilieva)Funda\u00E7\u00E3o para a Ci\u00EAncia e a Tecnologia under the projects UIDB/04564/2020 and UIDP/04564/2020 with DOI identifiers [http://dx.doi.org/10.54499/UIDB/04564/2020] and [http://dx.doi.org/10.54499/UIDP/04564/2020] (Rui D. M. Travasso);ICSC \u2014 Centro Nazionale di Ricerca in High Performance Computing, Big Data and Quantum Computing, funded by European Union \u2014 NextGenerationEU (Luca Tubiana, Raffaello Potestio)NSF-CLP under the projects CHE2145906 CAREER A Novel Twist in Nature; Proteins with a Pierced Lasso Topology (Ellinor Haglund)Funda\u00E7\u00E3o para a Ci\u00EAncia e a Tecnologia under the project UIDB/04046/2020[(https://doi.org/10.54499/UIDB/04046/2020)] (Patr\u00EDcia F. N. Fa\u00EDsca)Leverhulme Trust Grant under the project RP2013-K-017 (Dorothy Buck)Slovenian Research and Innovation Agency (ARIS) under the project P1-0099 Physics of soft matter, surfaces and nanostructures (Simon Copar, Miha Ravnik, Slobodan Zumer)Engineering and Physical Sciences Research Council (EPSRC) under the project EP/ T000961/1 (Anton Souslov)European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation Programme Grant Agreement 758588 (Raffaello Potestio).European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation Programme Grant Agreement 758588 (Raffaello Potestio). Finally, Luca Tubiana would like to acknowledge support from the Italian MIUR grant Rita Levi-Montalcini.

The last years have witnessed remarkable advances in our understanding of the emergence and consequences of topological constraints in biological and soft matter. Examples are abundant in relation to (bio)polymeric systems and range from the characterization of knots in single polymers and proteins to that of whole chromosomes and polymer melts. At the same time, considerable advances have been made in the description of the interplay between topological and physical properties in complex fluids, with the development of techniques that now allow researchers to control the formation of and interaction between defects in diverse classes of liquid crystals. Thanks to technological progress and the integration of experiments with increasingly sophisticated numerical simulations, topological biological and soft matter is a vibrant area of research attracting scientists from a broad range of disciplines. However, owing to the high degree of specialization of modern science, many results have remained confined to their own particular fields, with different jargon making it difficult for researchers to share ideas and work together towards a comprehensive view of the diverse phenomena at play. Compelled by these motivations, here we present a comprehensive overview of topological effects in systems ranging from DNA and genome organization to entangled proteins, polymeric materials, liquid crystals, and theoretical physics, with the intention of reducing the barriers between different fields of soft matter and biophysics. Particular care has been taken in providing a coherent formal introduction to the topological properties of polymers and of continuum materials and in highlighting the underlying common aspects concerning the emergence, characterization, and effects of topological objects in different systems. The second half of the review is dedicated to the presentation of the latest results in selected problems, specifically, the effects of topological constraints on the viscoelastic properties of polymeric materials; their relation with genome organization; a discussion on the emergence and possible effects of knots and other entanglements in proteins; the emergence and effects of topological defects and solitons in complex fluids. This review is dedicated to the memory of Marek Cieplak.