The paper presents a study of dark matter halo mass functions in Cold Dark Matter (CDM) cosmologies, combining data from multiple N-body simulations. The results show that the abundance of dark matter halos can be predicted with high accuracy across a wide range of masses, from $ \sim 3 \times 10^{11} h^{-1} M_{\odot} $ to $ \sim 5 \times 10^{15} h^{-1} M_{\odot} $. The simulations, including "Hubble Volume" simulations of $ \tau $ CDM and $ \Lambda $ CDM cosmologies, demonstrate that the mass function is nearly independent of cosmological parameters, redshift, and initial power spectrum when expressed in appropriate variables. This universality aligns with the predictions of the Press-Schechter model, although the model overestimates the abundance of "typical" halos and underestimates that of massive systems. The study compares the mass functions derived from different simulations and halo finding algorithms (friends-of-friends and spherical overdensity). It shows that the mass functions are consistent across different simulations and halo definitions, with systematic uncertainties below 10-30%. The results indicate that the mass function is robust and can be described by a single fitting formula, which is very close to the Sheth-Tormen model. This formula accurately predicts the mass functions for various CDM cosmologies, including $ \tau $ CDM, $ \Lambda $ CDM, SCDM, and OCDM, over a wide range of redshifts. The paper also discusses the sensitivity of the mass functions to numerical parameters such as particle mass, gravitational softening, and starting redshift. It shows that the mass functions are relatively insensitive to these parameters, with systematic uncertainties below 10-30%. The study concludes that the mass function is universal when halos are defined at fixed overdensity, independent of $ \Omega $, and that the Press-Schechter model is not accurate for predicting the mass function in all cases. The Sheth-Tormen model provides a better fit to the observed mass functions. The results highlight the importance of accurate theoretical predictions for halo mass functions in understanding the formation of galaxies and the evolution of dark matter halos in different cosmological models.The paper presents a study of dark matter halo mass functions in Cold Dark Matter (CDM) cosmologies, combining data from multiple N-body simulations. The results show that the abundance of dark matter halos can be predicted with high accuracy across a wide range of masses, from $ \sim 3 \times 10^{11} h^{-1} M_{\odot} $ to $ \sim 5 \times 10^{15} h^{-1} M_{\odot} $. The simulations, including "Hubble Volume" simulations of $ \tau $ CDM and $ \Lambda $ CDM cosmologies, demonstrate that the mass function is nearly independent of cosmological parameters, redshift, and initial power spectrum when expressed in appropriate variables. This universality aligns with the predictions of the Press-Schechter model, although the model overestimates the abundance of "typical" halos and underestimates that of massive systems. The study compares the mass functions derived from different simulations and halo finding algorithms (friends-of-friends and spherical overdensity). It shows that the mass functions are consistent across different simulations and halo definitions, with systematic uncertainties below 10-30%. The results indicate that the mass function is robust and can be described by a single fitting formula, which is very close to the Sheth-Tormen model. This formula accurately predicts the mass functions for various CDM cosmologies, including $ \tau $ CDM, $ \Lambda $ CDM, SCDM, and OCDM, over a wide range of redshifts. The paper also discusses the sensitivity of the mass functions to numerical parameters such as particle mass, gravitational softening, and starting redshift. It shows that the mass functions are relatively insensitive to these parameters, with systematic uncertainties below 10-30%. The study concludes that the mass function is universal when halos are defined at fixed overdensity, independent of $ \Omega $, and that the Press-Schechter model is not accurate for predicting the mass function in all cases. The Sheth-Tormen model provides a better fit to the observed mass functions. The results highlight the importance of accurate theoretical predictions for halo mass functions in understanding the formation of galaxies and the evolution of dark matter halos in different cosmological models.