The paper investigates the mass function of dark matter halos in a set of cosmological simulations of flat ΛCDM cosmology, focusing on its evolution at redshifts z ≤ 2. Halos are identified as isolated density peaks, and their masses are measured within specific overdensities. The study argues that spherical overdensity masses are more directly linked to cluster observables than masses measured using the friends-of-friends (FOF) algorithm, making them preferable for accurate forecasts of halo abundances. The simulation set allows calibration of the mass function at z = 0 for virial masses in the range 10¹¹ h⁻¹ M☉ ≤ M ≤ 10¹⁵ h⁻¹ M☉ to within ~5%. Fitting functions for the halo mass function are derived for a wide range of overdensities, both at z = 0 and earlier epochs. The main finding is that the mass function cannot be represented by a universal fitting function at this level of accuracy. The amplitude of the "universal" function decreases monotonically by ~20–50% from z = 0 to 2.5, depending on the mass definition. The study also finds evidence for redshift evolution in the overall shape of the mass function. The paper presents results for the mass function, demonstrating how it depends on cosmology and redshift. It discusses the comparison of FOF and SO halos, the effects of resolution, and the redshift evolution of the mass function. The results show that the mass function is not universal in redshift or for correspondingly large changes in cosmology at this level of accuracy. The study concludes that the mass function can be accurately calibrated using the spherical overdensity algorithm for a range of overdensities probed by observations and frequently used in theoretical calculations. The calibration for the standard ΛCDM cosmology is the main focus of the paper, with a focus on accurate calibration of halo abundances for intermediate and high-mass halos over the range of redshifts most relevant for current and upcoming large cluster surveys.The paper investigates the mass function of dark matter halos in a set of cosmological simulations of flat ΛCDM cosmology, focusing on its evolution at redshifts z ≤ 2. Halos are identified as isolated density peaks, and their masses are measured within specific overdensities. The study argues that spherical overdensity masses are more directly linked to cluster observables than masses measured using the friends-of-friends (FOF) algorithm, making them preferable for accurate forecasts of halo abundances. The simulation set allows calibration of the mass function at z = 0 for virial masses in the range 10¹¹ h⁻¹ M☉ ≤ M ≤ 10¹⁵ h⁻¹ M☉ to within ~5%. Fitting functions for the halo mass function are derived for a wide range of overdensities, both at z = 0 and earlier epochs. The main finding is that the mass function cannot be represented by a universal fitting function at this level of accuracy. The amplitude of the "universal" function decreases monotonically by ~20–50% from z = 0 to 2.5, depending on the mass definition. The study also finds evidence for redshift evolution in the overall shape of the mass function. The paper presents results for the mass function, demonstrating how it depends on cosmology and redshift. It discusses the comparison of FOF and SO halos, the effects of resolution, and the redshift evolution of the mass function. The results show that the mass function is not universal in redshift or for correspondingly large changes in cosmology at this level of accuracy. The study concludes that the mass function can be accurately calibrated using the spherical overdensity algorithm for a range of overdensities probed by observations and frequently used in theoretical calculations. The calibration for the standard ΛCDM cosmology is the main focus of the paper, with a focus on accurate calibration of halo abundances for intermediate and high-mass halos over the range of redshifts most relevant for current and upcoming large cluster surveys.