Lithium niobate on insulator (LNOI), by taking advantage of versatile properties of lithium niobate (LN) and a large refractive index contrast, provides an ideal on-chip platform for studying a broad range of optical effects as well as developing various superior photonic devices. It is a game-changer technology for traditional LN-based applications. Especially, with recent advances in the fabrication of high-quality micro-/nano-structures and devices on the LNOI platform, LN-based integrated photonics has been propelled to new heights. In this review, we summarize the latest research advances in lithium niobate thin film (LNTF), with a special focus on nonlinear wave mixing and their photonics applications. Different types of second- and third-order nonlinear processes in LNOI micro- and nano-structures are reviewed, including nonlinear frequency conversion, frequency comb generation and supercontinuum generation. Furthermore, perspectives for photonic integrated circuits (PICs) on the LNOI platform in nonlinear optics regime are predicted.