The eROSITA all-sky survey has provided new cosmological constraints from the abundance of galaxy clusters in the western Galactic hemisphere. Using data from the first eROSITA All-Sky Survey (eRASS1), researchers have analyzed 5259 galaxy clusters detected over an area of 12791 deg². The study combines data from eRASS1 with the Dark Energy Survey (DES), Kilo-Degree Survey (KiDS), and Hyper Supreme Camera (HSC) to calibrate the relationship between X-ray count rate and cluster mass through weak gravitational lensing measurements. The results constrain the ΛCDM parameters, including the matter density Ωm = 0.29 ± 0.02 and the normalization of density fluctuations σ8 = 0.88 ± 0.02, leading to S8 = 0.86 ± 0.01. These results are consistent with state-of-the-art cosmic microwave background (CMB) measurements. The study also places a stringent upper limit on the summed masses of left-handed light neutrinos to ∑mν < 0.43 eV (95% confidence interval) from cluster number counts alone. Combining eRASS1 data with CMB and ground-based neutrino oscillation experiments, the summed neutrino masses are measured as ∑mν = 0.09 ± 0.02 eV or 0.12 ± 0.03 eV, assuming normal or inverted mass hierarchy scenarios. The study significantly improves constraints on the dark energy equation of state parameter to w = -1.12 ± 0.12. The results from the first All-Sky Survey improve cosmological constraints by over a factor of 5 to 9 compared to previous cluster surveys, establishing cluster abundance measurements for precision cosmology and setting the stage for deeper eROSITA All-Sky Surveys and future cluster abundance experiments. The study also presents the selection function and Bayesian likelihood modeling method used in the analysis, as well as the validation process of the cosmology pipeline and blinding strategy. The results on scaling relations and various models for cosmology are presented, along with a discussion of systematics and an outlook for future eROSITA surveys. The study provides a detailed description of the galaxy cluster survey data, including the eRASS1 cluster catalog, the selection of the X-ray observable, and the weak-lensing follow-up observations and analysis. The study also presents the results from the Dark Energy Survey, the Hyper Suprime-Cam (HSC) survey, and the Kilo Degree Survey, including the shear extraction analysis and mass calibration strategy used in these surveys. The study concludes that the results are consistent with the concordance ΛCDM cosmology and that the eRASS1 cluster abundances significantly improve the constraints on the dark energy equation of state parameterThe eROSITA all-sky survey has provided new cosmological constraints from the abundance of galaxy clusters in the western Galactic hemisphere. Using data from the first eROSITA All-Sky Survey (eRASS1), researchers have analyzed 5259 galaxy clusters detected over an area of 12791 deg². The study combines data from eRASS1 with the Dark Energy Survey (DES), Kilo-Degree Survey (KiDS), and Hyper Supreme Camera (HSC) to calibrate the relationship between X-ray count rate and cluster mass through weak gravitational lensing measurements. The results constrain the ΛCDM parameters, including the matter density Ωm = 0.29 ± 0.02 and the normalization of density fluctuations σ8 = 0.88 ± 0.02, leading to S8 = 0.86 ± 0.01. These results are consistent with state-of-the-art cosmic microwave background (CMB) measurements. The study also places a stringent upper limit on the summed masses of left-handed light neutrinos to ∑mν < 0.43 eV (95% confidence interval) from cluster number counts alone. Combining eRASS1 data with CMB and ground-based neutrino oscillation experiments, the summed neutrino masses are measured as ∑mν = 0.09 ± 0.02 eV or 0.12 ± 0.03 eV, assuming normal or inverted mass hierarchy scenarios. The study significantly improves constraints on the dark energy equation of state parameter to w = -1.12 ± 0.12. The results from the first All-Sky Survey improve cosmological constraints by over a factor of 5 to 9 compared to previous cluster surveys, establishing cluster abundance measurements for precision cosmology and setting the stage for deeper eROSITA All-Sky Surveys and future cluster abundance experiments. The study also presents the selection function and Bayesian likelihood modeling method used in the analysis, as well as the validation process of the cosmology pipeline and blinding strategy. The results on scaling relations and various models for cosmology are presented, along with a discussion of systematics and an outlook for future eROSITA surveys. The study provides a detailed description of the galaxy cluster survey data, including the eRASS1 cluster catalog, the selection of the X-ray observable, and the weak-lensing follow-up observations and analysis. The study also presents the results from the Dark Energy Survey, the Hyper Suprime-Cam (HSC) survey, and the Kilo Degree Survey, including the shear extraction analysis and mass calibration strategy used in these surveys. The study concludes that the results are consistent with the concordance ΛCDM cosmology and that the eRASS1 cluster abundances significantly improve the constraints on the dark energy equation of state parameter