1. Introduction  
2. Galaxy cluster survey data  
2.1. eRASS1 Cluster Catalog  
2.2. Selection of the X-ray observable  
2.3. Weak-lensing follow-up observations and analysis  
3. The X-ray selection function  
3.1. Simulations  
3.2. Event generation and catalog inference  
3.4. Selection models for validation and blinding  
3.5. Optical selection function  
4. Likelihood  
4.1. Global likelihood function  
4.2. Mixture method for the contamination modeling  
4.3. Cluster number counts  
4.4. Weak-lensing mass calibration likelihood  
4.5. Optical likelihood  
4.6. Parameters and priors  
5. Validation of the cosmology pipeline and the blinding strategy  
5.1. Mocks generation and validation of the pipeline  
5.2. Blinding strategy  
6. Scaling relations  
6.1. X-ray Count Rate - Mass Scaling Relation  
6.2. Weak-lensing inferred mass-mass scaling relation  
6.3. Richness-mass scaling relation  
7. Cosmological constraints  
7.1. Goodness of fit  
7.2. Constraints on the Λ CDM cosmology  
7.3. Constraints on the ν CDM cosmology  
7.4. Constraints on the $w$ CDM cosmology  
7.5. Constraints on the ν w CDM cosmology  
7.6. Model comparison  
8. Discussion  
8.1. Methodology  
8.2. $S_{8}$ Measurements  
8.3. Constraints on the total mass of neutrinos  
8.4. Constraints on the dark energy equation of state  
8.5. Future improvements  
9. Summary