Photonics on thin-film lithium niobate (TFLN) has emerged as one of the most pursued disciplines within integrated optics. Ultracompact and low-loss optical waveguides and related devices on this modern material platform have rejuvenated the traditional and commercial applications of lithium niobate for optical modulators based on the electro-optic effect, as well as optical wavelength converters based on second-order nonlinear effects, e.g., second-harmonic, sum-, and difference-frequency generations. TFLN has also created vast opportunities for applications and integrated solutions for optical parametric amplification and oscillation, cascaded nonlinear effects, such as low-harmonic generation; third-order nonlinear effects, such as supercontinuum generation; optical frequency comb generation and stabilization; and nonclassical nonlinear effects, such as spontaneous parametric downconversion for quantum optics. Recent progress in nonlinear integrated photonics on TFLN for all these applications, their current trends, and future opportunities and challenges are reviewed.