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Full list of authors: Amaro-Seoane, Pau; Andrews, Jeff; Sedda, Manuel Arca; Askar, Abbas.; Baghi, Quentin; Balasov, Razvan; Bartos, Imre; Bavera, Simone S.; Bellovary, Jillian; Berry, Christopher P. L.; Berti, Emanuele; Bianchi, Stefano; Blecha, Laura; Blondin, Stephane; Bogdanovic, Tamara; Boissier, Samuel; Bonetti, Matteo; Bonoli, Silvia; Bortolas, Elisa; Breivik, Katelyn; Capelo, Pedro R.; Caramete, Laurentiu; Cattorini, Federico; Charisi, Maria; Chaty, Sylvain; Chen, Xian; Chruslinska, Martyna; Chua, Alvin J. K.; Church, Ross; Colpi, Monica; D'Orazio, Daniel; Danielski, Camilla; Davies, Melvyn B.; Dayal, Pratika; De Rosa, Alessandra; Derdzinski, Andrea; Destounis, Kyriakos; Dotti, Massimo; Dutan, Ioana; Dvorkin, Irina; Fabj, Gaia; Foglizzo, Thierry; Ford, Saavik; Fouvry, Jean-Baptiste; Franchini, Alessia; Fragos, Tassos; Fryer, Chris; Gaspari, Massimo; Gerosa, Davide; Graziani, Luca; Groot, Paul; Habouzit, Melanie; Haggard, Daryl; Haiman, Zoltan; Han, Wen-Biao; Istrate, Alina; Johansson, Peter H.; Khan, Fazeel Mahmood; Kimpson, Tomas; Kokkotas, Kostas; Kong, Albert; Korol, Valeriya; Kremer, Kyle; Kupfer, Thomas; Lamberts, Astrid; Larson, Shane; Lau, Mike; Liu, Dongliang; Lloyd-Ronning, Nicole; Lodato, Giuseppe; Lupi, Alessandro; Ma, Chung-Pei; Maccarone, Tomas; Mandel, Ilya; Mangiagli, Alberto; Mapelli, Michela; Mathis, Stephane; Mayer, Lucio; McGee, Sean; McKernan, Berry; Miller, M. Coleman; Mota, David F.; Mumpower, Matthew; Nasim, Syeda S.; Nelemans, Gijs; Noble, Scott; Pacucci, Fabio; Panessa, Francesca; Paschalidis, Vasileios; Pfister, Hugo; Porquet, Delphine; Quenby, John; Ricarte, Angelo; Roepke, Friedrich K.; Regan, John; Rosswog, Stephan; Ruiter, Ashley; Ruiz, Milton; Runnoe, Jessie; Schneider, Raffaella; Schnittman, Jeremy; Secunda, Amy; Sesana, Alberto; Seto, Naoki; Shao, Lijing; Shapiro, Stuart; Sopuerta, Carlos; Stone, Nicholas C.; Suvorov, Arthur; Tamanini, Nicola; Tamfal, Tomas; Tauris, Thomas; Temmink, Karel; Tomsick, John; Toonen, Silvia; Torres-Orjuela, Alejandro; Toscani, Martina; Tsokaros, Antonios; Unal, Caner; Vazquez-Aceves, Veronica; Valiante, Rosa; van Putten, Maurice; van Roestel, Jan; Vignali, Christian; Volonteri, Marta; Wu, Kinwah; Younsi, Ziri; Yu, Shenghua; Zane, Silvia; Zwick, Lorenz; Antonini, Fabio; Baibhav, Vishal; Barausse, Enrico; Bonilla Rivera, Alexander; Branchesi, Marica; Branduardi-Raymont, Graziella; Burdge, Kevin; Chakraborty, Srija; Cuadra, Jorge; Dage, Kristen; Davis, Benjamin; de Mink, Selma E.; Decarli, Roberto; Doneva, Daniela; Escoffier, Stephanie; Gandhi, Poshak; Haardt, Francesco; Lousto, Carlos O.; Nissanke, Samaya; Nordhaus, Jason; O'Shaughnessy, Richard; Portegies Zwart, Simon; Pound, Adam; Schussler, Fabian; Sergijenko, Olga; Spallicci, Alessandro; Vernieri, Daniele; Vigna-Gomez, Alejandro.-- This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/., The Laser Interferometer Space Antenna (LISA) will be a transformative experiment for gravitational wave astronomy, and, as such, it will offer unique opportunities to address many key astrophysical questions in a completely novel way. The synergy with ground-based and space-born instruments in the electromagnetic domain, by enabling multi-messenger observations, will add further to the discovery potential of LISA. The next decade is crucial to prepare the astrophysical community for LISA’s first observations. This review outlines the extensive landscape of astrophysical theory, numerical simulations, and astronomical observations that are instrumental for modeling and interpreting the upcoming LISA datastream. To this aim, the current knowledge in three main source classes for LISA is reviewed; ultra-compact stellar-mass binaries, massive black hole binaries, and extreme or interme-diate mass ratio inspirals. The relevant astrophysical processes and the established modeling techniques are summarized. Likewise, open issues and gaps in our understanding of these sources are highlighted, along with an indication of how LISA could help making progress in the different areas. New research avenues that LISA itself, or its joint exploitation with upcoming studies in the electromagnetic domain, will enable, are also illustrated. Improvements in modeling and analysis approaches, such as the combination of numerical simulations and modern data science techniques, are discussed. This review is intended to be a starting point for using LISA as a new discovery tool for understanding our Universe. © The Author(s) 2023, corrected publication 2023., P. Dayal acknowledges support from the European Research council (ERC-717001) and from the Netherlands Research Council NWO (016.VIDI.189.162). P.H. Johansson acknowledges the support from the European Research Council (ERC-818930). S. Toonen acknowledges support from the Netherlands Research Council NWO (VENI 639.041.645 Grants) C. Unal is supported by European Structural and Investment Funds and the Czech Ministry of Education, Youth and Sports (Project CoGraDS - CZ.02.1.01/0.0/0.0/15_003/0000437). S. Chaty acknowledges the LabEx UnivEarthS for the funding of Interface project I10 “From binary evolution towards merging of compact objects”. A. De Rosa acknowledges financial contribution from the agreement ASI-INAF n.2017-14-H.O E. Berti is supported by NSF Grants No. PHY-1912550 and AST-2006538, NASA ATP Grants No. 17-ATP17-0225 and 19-ATP19-0051, NSF-XSEDE Grant No. PHY-090003, and NSF Grant PHY-20043. D. Gerosa is supported by European Union’s H2020 ERC Starting Grant No. 945155–GWmining, Leverhulme Trust Grant No. RPG-2019-350 and Royal Society Grant No. RGS-R2-202004. T. Bogdanovic acknowledges support by the NASA award No. 80NSSC19K0319 and by the NSF award AST-1908042. D. Porquet acknowledges funding support from CNES. C. Danielski acknowledges financial support from the State Agency for Research of the Spanish MCIU through the “Center of Excellence Severo Ochoa” award to the Instituto de Astrofísica de Andalucía (SEV-2017-0709) B.L. Davis acknowledges support from Tamkeen under the NYU Abu Dhabi Research Institute Grant CAP3. F. Pacucci acknowledges support from a Clay Fellowship by the SAO and from the Black Hole Initiative, which is funded by grants from the John Templeton Foundation and the Gordon and Betty Moore Foundation. A.J. Ruiter acknowledges support from the Australian Research Council Future Fellowship Grant FT170100243. V. Paschalidis is supported by NSF Grant PHY-1912619 and NASA Grant 80NSSC20K1542 to the University of Arizona, and NSF-XSEDE Grant TG-PHY190020. D. Haggard acknowledges support from the NSERC Discovery Grant and Canada Research Chairs programs, and the Bob Wares Science Innovation Prospectors Fund. M. Toscani acknowledges European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Grant Agreement No. 823823 (RISE DUSTBUSTERS project) and COST Action CA16104 - Gravitational waves, black holes and fundamental physics, supported by COST (European Cooperation in Science and Technology). M. Chruslinska, A. Istrate and G. Nelemans acknowledge support from Netherlands Research Council NWO. T. Fragos and S. Bavera acknowledge support from a Swiss National Science Foundation Professorship Grant (project numbers PP00P2_176868 and PP00P2_211006)., With funding from the Spanish government through the "Severo Ochoa Centre of Excellence" accreditation (CEX2021-001131-S).

Tidal Disruption Events (TDEs) occur when a star, wandering too close to a black hole (BH), is torn apart by BH tides. After the disruption, roughly half of the stellar debris is expected to circularize around the BH and eventually form an accretion disc. These events represent a unique way to detect BHs through the universe since they emit both electromagnetic (EM) radiation and gravitational waves (GWs). In fact, after returning to the pericenter, the stellar material accretes onto the central object, producing very luminous EM flares. To date, around 50 robust TDEs have been detected in different bands of the EM spectrum (optical, X-ray, radio). Furthermore, these events also produce GWs during the whole process: while the star is stretched and compressed by BH tides, when the star is disrupted at pericenter and during the circularization process. So far, these gravitational signals have been inaccessible to us since they are low-frequency signals. Hence, these events will be seen by the future space-based detectors (like LISA and deci-Hertz observatories). In the upcoming future, thanks to the synergy between the spatial GW observatories and new and more performant telescopes, we will observe these events both via GWs and EM radiation. This will provide us a unique and more complete way to study BHs. Given these premises, this Thesis aims to study the main features of TDE gravitational emission, both through analytical and numerical studies. In particular, I explore the signal from an individual TDE, focussing on the emission generated at later stages during the circularization of the debris. Then, I investigate the signal produced by the entire cosmic population of tidal disruptions. Finally, I illustrate a catalogue of gravitational waveforms from TDEs built with a numerical tool implemented by myself during the PhD.