In this work we propose an interface treatment for conjugate heat and mass transfer with discontinuities or jumps of temperature (concentration) and/or heat (mass) flux at the interface using the lattice Boltzmann equation (LBE) method. The present interface treatment is based on the second-order accurate boundary condition treatments for Dirichlet and Neumann problems (Li et al., 2013) and second-order accurate interface treatment for standard conjugate heat and mass transfer with the continuity of temperature (concentration) and flux at the interface (Li et al., 2014). The interfacial jump conditions are intrinsically satisfied in the present treatment without iterative computations that are typically needed in conventional finite-difference or finite-volume methods. The interfacial temperature (concentration) values and the fluxes into the adjacent domains are conveniently obtained from the microscopic distribution functions in the LBE model without finite-difference approximations. Since the local intersection link fraction is included in the present treatment, the interfacial geometry is preserved and the present interface schemes are capable of handling curved interfaces. The numerical accuracy and convergence of the present interface schemes are verified with several numerical tests, including (i) one-dimensional (1D) steady diffusion within a two-solid slab; the slab is either aligned with the lattice velocity vector or with an inclination angle, (ii) 2D steady diffusion in a circular domain of two concentric solids, (iii) 3D steady diffusion in a spherical domain of two concentric solids, and (iv) 2D steady convection-diffusion in a channel. The two adjacent domains have different thermal (mass) transport properties and specific temperature (concentration) and flux jump conditions are imposed at the interface in each of those tests. It is verified that the present interface treatment for jump conditions does not introduce additional errors compared to the case without jumps; and second-order accuracy in space is obtained for the interior temperature (concentration) field, the interfacial temperature (concentration) values and interfacial fluxes for straight interfaces aligned with the lattice velocity vector in both diffusion and convection-diffusion problems. The effects of inclined and curved interfacial geometry on the order-of-accuracy of the LBE results are also investigated and the results are compared with previous findings. (C) 2015 Elsevier Ltd. All rights reserved.