Crystalline lithium niobate (LN) is an important optical material because of its broad transmission window that spans from ultraviolet to mid-infrared and its large nonlinear and electro-optic coefficients. Furthermore, the recent development and commercialization of LN-on-insulator (LNOI) technology has opened an avenue for the realization of integrated on-chip photonic devices with unprecedented performances in terms of propagation loss, optical nonlinearity, and electro-optic tunability. This review begins with a brief introduction of the history and current status of LNOI photonics. We then discuss the fabrication techniques of LNOI-based photonic structures and devices. The recent revolution in the LN photonic industry has been sparked and is still being powered by innovations of the nanofabrication technology of LNOI, which enables the production of building block structures, such as optical microresonators and waveguides of unprecedented optical qualities. The following sections present various on-chip LNOI devices categorized into nonlinear photonic and electro-optic tunable devices and photonic-integrated circuits. Some conclusions and future perspectives are provided.