We will discuss the construction of arbitrarily high order accurate schemes which satisfy a strict maximum principle for nonlinear scalar conservation laws, passive convection in incompressible flows, and nonlinear scalar convection-diffusion equations, and preserve positivity for density and pressure for compressible Euler systems in gas dynamics. Take Euler systems for an example, the L1-stability of mass and energy can be achieved by enforcing the positivity of density and pressure during the time evolution. The main difficulty is how to enforce the positivity without destroying the high order accuracy and the local conservation, which was unknown previously for higher than second order accurate schemes. For discontinuous Galerkin (DG) and weighted essentially non-oscillatory (WENO) finite volume (FV) and finite difference (FD) schemes, a general framework (for any order of accuracy and unstructured meshes in any dimension) is established to construct a simple scaling limiter involving only the polynomial solution at certain quadrature points. This is the first time that a genuinely high order positivity-preserving scheme is constructed in high dimensions. The additional computational cost due to the limiter is marginal and the limiter does not affect the parallelizability of the base scheme. Some recent extensions will be discussed. This part is a joint work with Chi-Wang Shu at Brown University.