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International audience; The study of nonlinear optics in the 1960s attracted considerable attention from a theoretical standpoint, engendering many proposals for practical use of these new photonic effects. Among these suggestions, the development of efficient two-photon absorption (2PA) has attracted sustained interest due to its demonstrated (or potential) use in a broad range of applications that include optical data storage, optical limiting and nanofabrication. The use of 2PA in biological applications is particularly appealing. This is because 2PA offers several advantages for biooriented applications, such as intrinsic three-dimensional resolution, increased penetration depth in biological materials such as tissue and highly focused excitation at half-energy, leading to a decrease of auto-fluorescence and photodamage. In this Primer Review, we introduce the essential background theory needed for an understanding of the field, we describe the key experiments deployed to quantify material performance, we discuss the evolution of 2PA molecular design, and we summarise the state-of-the-art and the existing challenges in the use of 2PA in imaging, therapy and theranostics.