A new X-ray selected and flux-limited galaxy cluster sample is presented, based on the ROSAT All-Sky Survey. The sample includes 63 brightest clusters with galactic latitude |b_II| ≥ 20 deg and X-ray flux f_X(0.1–2.4 keV) ≥ 2 × 10⁻¹¹ erg s⁻¹ cm⁻². Gravitational masses are determined using intracluster gas density profiles and gas temperatures, assuming hydrostatic equilibrium. The sample is used to establish the X-ray luminosity-gravitational mass relation and determine the cluster mass function. The mass function is used to constrain the mean cosmic matter density and the amplitude of mass fluctuations. Comparison to Press-Schechter type models in Cold Dark Matter cosmological models yields constraints Ω_m = 0.12₋₀.₀₄⁺₀.₀₆ and σ₈ = 0.96₋₀.₁₂⁺₀.₁₅ (90% c.l.). Systematic uncertainties are quantified, and the mass function is integrated to show that the contribution of mass bound within virialized cluster regions to the total matter density is small, Ω_Cluster = 0.012₋₀.₀₀₄⁺₀.₀₀₃ for cluster masses larger than 6.4₋₀.₆⁺₀.₇ × 10¹³ h₅₀⁻¹ M☉. The mass function is used to test cosmological models and determine the total matter density in the universe. The sample is constructed from multiple catalogs, including REFLEX, NORAS, NORAS II, BCS, RASS 1 Bright Sample, XBACs, and others. The sample is highly complete and homogeneous, with a flux limit of 2.0 × 10⁻¹¹ erg s⁻¹ cm⁻². The mass function is determined using the V_max method, with different approaches to account for scatter in the L_X-M_tot relation. The mass function is compared to previous determinations and predictions of cosmological models, showing that the contribution of galaxy clusters to the total matter density is small. The results are used to constrain cosmological parameters and test cosmological models.A new X-ray selected and flux-limited galaxy cluster sample is presented, based on the ROSAT All-Sky Survey. The sample includes 63 brightest clusters with galactic latitude |b_II| ≥ 20 deg and X-ray flux f_X(0.1–2.4 keV) ≥ 2 × 10⁻¹¹ erg s⁻¹ cm⁻². Gravitational masses are determined using intracluster gas density profiles and gas temperatures, assuming hydrostatic equilibrium. The sample is used to establish the X-ray luminosity-gravitational mass relation and determine the cluster mass function. The mass function is used to constrain the mean cosmic matter density and the amplitude of mass fluctuations. Comparison to Press-Schechter type models in Cold Dark Matter cosmological models yields constraints Ω_m = 0.12₋₀.₀₄⁺₀.₀₆ and σ₈ = 0.96₋₀.₁₂⁺₀.₁₅ (90% c.l.). Systematic uncertainties are quantified, and the mass function is integrated to show that the contribution of mass bound within virialized cluster regions to the total matter density is small, Ω_Cluster = 0.012₋₀.₀₀₄⁺₀.₀₀₃ for cluster masses larger than 6.4₋₀.₆⁺₀.₇ × 10¹³ h₅₀⁻¹ M☉. The mass function is used to test cosmological models and determine the total matter density in the universe. The sample is constructed from multiple catalogs, including REFLEX, NORAS, NORAS II, BCS, RASS 1 Bright Sample, XBACs, and others. The sample is highly complete and homogeneous, with a flux limit of 2.0 × 10⁻¹¹ erg s⁻¹ cm⁻². The mass function is determined using the V_max method, with different approaches to account for scatter in the L_X-M_tot relation. The mass function is compared to previous determinations and predictions of cosmological models, showing that the contribution of galaxy clusters to the total matter density is small. The results are used to constrain cosmological parameters and test cosmological models.