The random-energy model (REM) is introduced as a simple, exactly solvable model of disordered systems. The model describes a system with $2^N$ energy levels, which are independent random variables distributed according to a specific probability distribution. The REM exhibits a phase transition at a critical temperature $T_c$, and the low-temperature phase is completely frozen. The model's thermodynamic properties, such as the specific heat and ground-state energy, are calculated in detail. The REM's behavior is compared with that of the Sherrington-Kirkpatrick (SK) model, showing qualitative similarities. The replica method is analyzed, and it is shown that an unstable saddle point gives the correct low-temperature free energy. The REM's behavior in the presence of a uniform magnetic field and ferromagnetic pair interactions is studied, leading to phase diagrams similar to those of the SK model. The REM provides lower bounds for the ground-state energies of a large class of spin-glass models. The paper concludes by discussing the limitations of the replica method and the need to account for correlations between energy levels in more realistic models.The random-energy model (REM) is introduced as a simple, exactly solvable model of disordered systems. The model describes a system with $2^N$ energy levels, which are independent random variables distributed according to a specific probability distribution. The REM exhibits a phase transition at a critical temperature $T_c$, and the low-temperature phase is completely frozen. The model's thermodynamic properties, such as the specific heat and ground-state energy, are calculated in detail. The REM's behavior is compared with that of the Sherrington-Kirkpatrick (SK) model, showing qualitative similarities. The replica method is analyzed, and it is shown that an unstable saddle point gives the correct low-temperature free energy. The REM's behavior in the presence of a uniform magnetic field and ferromagnetic pair interactions is studied, leading to phase diagrams similar to those of the SK model. The REM provides lower bounds for the ground-state energies of a large class of spin-glass models. The paper concludes by discussing the limitations of the replica method and the need to account for correlations between energy levels in more realistic models.