Spin glasses are magnetic alloys where magnetic moments (spins) exhibit non-periodic freezing, slow response, and low-temperature heat capacity similar to conventional glasses. The study of spin glasses has led to new analytical, experimental, and computational techniques, revealing complex cooperative behavior in systems with competing interactions. Spin glass behavior arises from quenched disorder and frustration, where interactions conflict, preventing all from being satisfied simultaneously. The random-bond Ising model is a key example, with Hamiltonian $ H = -\sum_{(ij)} J_{ij} \sigma_i \sigma_j $, where $ J_{ij} $ are random interactions. Analytic theory, particularly replica theory, has been used to analyze spin glasses. Replica theory involves considering multiple copies of the system and averaging over disorder. The replica-symmetric ansatz assumes all replicas are equivalent, but this is unstable at low temperatures, leading to replica symmetry breaking (RSB). Parisi's ansatz, which introduces a continuous order parameter $ q(x) $, describes the free energy landscape as having hierarchical, ultrametric structures. This leads to non-self-averaging of overlaps and complex, chaotic free energy landscapes. The Sherrington-Kirkpatrick (SK) model, where each spin interacts with every other, is a special case where mean-field theory is exact. The SK model exhibits a phase transition at $ T_g = J/k $, with a critical temperature where the susceptibility shows a cusp. The Parisi ansatz predicts ultrametricity and non-self-averaging, with the overlap distribution showing hierarchical structures. The SK model's free energy landscape is chaotic, with small changes in parameters leading to significant changes in the hill-valley structure. Other infinite-ranged models are analyzed using similar techniques, leading to mean-field solutions. The study of spin glasses has broad implications, from condensed matter physics to biology and social systems, highlighting the importance of disorder and competition in complex systems.Spin glasses are magnetic alloys where magnetic moments (spins) exhibit non-periodic freezing, slow response, and low-temperature heat capacity similar to conventional glasses. The study of spin glasses has led to new analytical, experimental, and computational techniques, revealing complex cooperative behavior in systems with competing interactions. Spin glass behavior arises from quenched disorder and frustration, where interactions conflict, preventing all from being satisfied simultaneously. The random-bond Ising model is a key example, with Hamiltonian $ H = -\sum_{(ij)} J_{ij} \sigma_i \sigma_j $, where $ J_{ij} $ are random interactions. Analytic theory, particularly replica theory, has been used to analyze spin glasses. Replica theory involves considering multiple copies of the system and averaging over disorder. The replica-symmetric ansatz assumes all replicas are equivalent, but this is unstable at low temperatures, leading to replica symmetry breaking (RSB). Parisi's ansatz, which introduces a continuous order parameter $ q(x) $, describes the free energy landscape as having hierarchical, ultrametric structures. This leads to non-self-averaging of overlaps and complex, chaotic free energy landscapes. The Sherrington-Kirkpatrick (SK) model, where each spin interacts with every other, is a special case where mean-field theory is exact. The SK model exhibits a phase transition at $ T_g = J/k $, with a critical temperature where the susceptibility shows a cusp. The Parisi ansatz predicts ultrametricity and non-self-averaging, with the overlap distribution showing hierarchical structures. The SK model's free energy landscape is chaotic, with small changes in parameters leading to significant changes in the hill-valley structure. Other infinite-ranged models are analyzed using similar techniques, leading to mean-field solutions. The study of spin glasses has broad implications, from condensed matter physics to biology and social systems, highlighting the importance of disorder and competition in complex systems.