This paper introduces a class of numerical schemes, called Monotonicity Preserving Weighted Essentially Non-oscillatory (MPWENO) schemes, which are higher-order extensions of the weighted essentially non-oscillatory (WENO) schemes. These schemes are designed to maintain monotonicity while achieving high phase accuracy and order of accuracy. The MPWENO schemes are stable under standard CFL numbers and are efficient with computational complexity similar to lower-order WENO schemes. They are particularly useful for problems containing both discontinuities and complex smooth structures, such as compressible turbulence. The paper also explores the role of steepening algorithms, such as the artificial compression method, in preserving the order of accuracy for practical problems. Extensive testing is conducted to evaluate the performance of the MPWENO schemes, including convergence tests and multidimensional tests, demonstrating their effectiveness in various scenarios.This paper introduces a class of numerical schemes, called Monotonicity Preserving Weighted Essentially Non-oscillatory (MPWENO) schemes, which are higher-order extensions of the weighted essentially non-oscillatory (WENO) schemes. These schemes are designed to maintain monotonicity while achieving high phase accuracy and order of accuracy. The MPWENO schemes are stable under standard CFL numbers and are efficient with computational complexity similar to lower-order WENO schemes. They are particularly useful for problems containing both discontinuities and complex smooth structures, such as compressible turbulence. The paper also explores the role of steepening algorithms, such as the artificial compression method, in preserving the order of accuracy for practical problems. Extensive testing is conducted to evaluate the performance of the MPWENO schemes, including convergence tests and multidimensional tests, demonstrating their effectiveness in various scenarios.