Large two-photon absorption (2PA) cross-section combined with high emission quantum efficiency and size-tunable bandgap energy has put colloidal semiconductor nanocrystals (NCs) on the vanguard of nonlinear optical materials. After nearly two decades of intense studies on the nonlinear optical response in quantum-confined semiconductors, this is still a vibrant field, as novel nanomaterials are being developed and new applications are being proposed. In this review, we examine the progress of 2PA research in NCs, highlighting the impact of quantum confinement on the magnitude and spectral characteristics of this nonlinear response in semiconductor materials. We show that for NCs with three-dimensional quantum confinement, the so-called quantum dots, 2PA cross-section grows linearly with the nanoparticle volume, following a universal volume scaling. We overview strategies used to gain further control over the nonlinear optical response in these structures by shape and heterostructure engineering and some applications that might take advantage of the series of unique properties of these nanostructures.
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