The random-energy model is an exactly solvable model of disordered systems. Introduced by Bernard Derrida, it describes a system with energy levels that are independent random variables. The model is the limit of a family of disordered models when correlations between energy levels become negligible. It exhibits a phase transition, with a completely frozen low-temperature phase. The model's thermodynamic properties, including magnetic susceptibility and phase diagrams, are analyzed. The random-energy model shares many qualitative features with the Sherrington-Kirkpatrick (SK) model, but it provides lower bounds for the ground-state energy of a wide class of spin-glass models. The replica method is discussed, and it is shown that an unstable saddle point gives the low-temperature free energy. The model is extended to include magnetic fields and ferromagnetic pair interactions, with similar phase diagrams to the SK model. The model's behavior in the thermodynamic limit is studied, and it is shown that the infinite temperature is an accumulation point of critical temperatures. The model's properties are compared to those of the SK model, and it is concluded that the random-energy model provides a useful framework for understanding disordered systems. The paper also discusses the limitations of the replica method and the importance of considering energy-level correlations in realistic spin-glass models.The random-energy model is an exactly solvable model of disordered systems. Introduced by Bernard Derrida, it describes a system with energy levels that are independent random variables. The model is the limit of a family of disordered models when correlations between energy levels become negligible. It exhibits a phase transition, with a completely frozen low-temperature phase. The model's thermodynamic properties, including magnetic susceptibility and phase diagrams, are analyzed. The random-energy model shares many qualitative features with the Sherrington-Kirkpatrick (SK) model, but it provides lower bounds for the ground-state energy of a wide class of spin-glass models. The replica method is discussed, and it is shown that an unstable saddle point gives the low-temperature free energy. The model is extended to include magnetic fields and ferromagnetic pair interactions, with similar phase diagrams to the SK model. The model's behavior in the thermodynamic limit is studied, and it is shown that the infinite temperature is an accumulation point of critical temperatures. The model's properties are compared to those of the SK model, and it is concluded that the random-energy model provides a useful framework for understanding disordered systems. The paper also discusses the limitations of the replica method and the importance of considering energy-level correlations in realistic spin-glass models.