In this talk, I will present the convergence analysis and error estimate of nonlinear positivity-preserving finite volume schemes for diffusion problems. These schemes are also named by nonlinear two-point flux approximation (NTPFA) methods. Due to their conservation and monotonicity, such schemes have attracted wide attention. However, the convergence analysis of them is still not available. In this work, we first give the convergence of nonlinear positivity-preserving finite volume methods providing the uniform coercivity of these numerical schemes, and then prove an $H^{1}$ error estimate through the analysis of consistent errors under suitable regularity assumptions on the exact solution. Some numerical examples are presented to demonstrate the theoretical results.