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Thin-film lithium niobate on insulator (LNOI) is a promising platform for optical communications, microwave photonics, and quantum technologies. While many high-performance devices like electro-optic modulators and frequency comb sources have been achieved on LNOI platform, it remains challenging to realize photodetectors (PDs) on LNOI platform using simple and low-cost fabrication techniques. Two-dimensional (2D) materials are excellent candidates to achieve photodetection since they feature strong light-matter interaction, excellent mechanical flexibility, and potential large-scale complementary metal-oxide-semiconductor-compatible fabrication. In this work, we propose to address this demand using an LNOI-2D material platform and demonstrate two types of high-performance LNOI waveguide-integrated 2D material PDs, namely graphene and Tellurium (Te). Specifically, the LNOI-graphene PD features broadband operations at telecom and visible wavelengths, high normalized photocurrent-to-dark current ratios up to 3*106 W-1, and large 3-dB photoelectric bandwidths of over 40 GHz, simultaneously. The LNOI-Te PD on the other hand provides an ultrahigh responsivity of 7 A/W under 0.5 V bias for telecom optical signals while supporting GHz frequency responses. Our results show that the versatile properties of 2D materials and their excellent compatibility with LNOI waveguides could provide important low-cost solutions for system operating point monitoring and high-speed photoelectric conversion in future LN photonic integrated circuits.