This paper presents a class of numerical schemes that are higher-order extensions of the weighted essentially non-oscillatory (WENO) schemes. These schemes, called monotonicity preserving WENO (MPWENO) schemes, are designed to maintain monotonicity while achieving high-order accuracy. The schemes are stable under normal Courant numbers and are efficient, with computational complexity not significantly greater than that of lower-order schemes. They are particularly useful for problems containing both discontinuities and smooth regions, making them viable competitors to older total variation diminishing (TVD) schemes. The MPWENO schemes are shown to have high phase accuracy and high-order accuracy, with higher-order members being almost spectrally accurate for smooth problems. The schemes are also robust in capturing shocks and are capable of maintaining accuracy even with high-order extensions. The paper also discusses the role of steepening algorithms like the artificial compression method in the design of high-order schemes and presents several test problems in one and two dimensions. The results show that the MPWENO schemes have a substantial advantage over lower-order schemes for multidimensional problems where the flow is not aligned with the grid directions. The methodology is applicable to other hyperbolic systems, including magnetohydrodynamics, demonstrating the effectiveness of the MPWENO schemes in various contexts. The paper also explores the use of monotonicity preserving bounds to ensure the schemes remain monotonicity preserving while maintaining high-order accuracy. The results indicate that the MPWENO schemes are highly accurate and efficient, making them suitable for a wide range of applications in computational fluid dynamics.This paper presents a class of numerical schemes that are higher-order extensions of the weighted essentially non-oscillatory (WENO) schemes. These schemes, called monotonicity preserving WENO (MPWENO) schemes, are designed to maintain monotonicity while achieving high-order accuracy. The schemes are stable under normal Courant numbers and are efficient, with computational complexity not significantly greater than that of lower-order schemes. They are particularly useful for problems containing both discontinuities and smooth regions, making them viable competitors to older total variation diminishing (TVD) schemes. The MPWENO schemes are shown to have high phase accuracy and high-order accuracy, with higher-order members being almost spectrally accurate for smooth problems. The schemes are also robust in capturing shocks and are capable of maintaining accuracy even with high-order extensions. The paper also discusses the role of steepening algorithms like the artificial compression method in the design of high-order schemes and presents several test problems in one and two dimensions. The results show that the MPWENO schemes have a substantial advantage over lower-order schemes for multidimensional problems where the flow is not aligned with the grid directions. The methodology is applicable to other hyperbolic systems, including magnetohydrodynamics, demonstrating the effectiveness of the MPWENO schemes in various contexts. The paper also explores the use of monotonicity preserving bounds to ensure the schemes remain monotonicity preserving while maintaining high-order accuracy. The results indicate that the MPWENO schemes are highly accurate and efficient, making them suitable for a wide range of applications in computational fluid dynamics.