Black phosphorus (BP) is a recently rediscovered layered two-dimensional (2D) semiconductor with a direct band gap (0.35-2 eV), high hole mobility (300-5000 cm2/Vs), and transport anisotropy. In this paper, we systematically investigated the effects of metal-semiconductor interface/contacts on the performance of BP Schottky barrier transistors. First, a "clean" metal-BP contact is formed with boron nitride (BN) passivation. It is found that the contact resistance of the clean metal-BP contact is seven times less than the previously reported values. As a result, high-performance top-gate BP transistors show a record high ON-state drain current (I on) of 940 μA/μm. Second, BN tunneling barriers are formed at the source/drain contacts to help understand the abnormally high OFF-state drain current (I off) in devices with clean metal-BP contacts. This high I off is attributed to the electron tunneling current from the drain to the channel. Finally, the I on/I off of BP field-effect transistors can be significantly improved by using an asymmetric contact structure. By inserting a thin BN tunneling barrier at the drain side, I off is reduced by a factor of ∼120 with a cost of 20% reduction in I on. This case study of contacts on BP reveals the importance of understanding the metal-semiconductor contacts for 2D Schottky barrier transistors in general.