Solid-rotor induction machines have gained significant attention in various industrial applications due to their robustness, reliability, and cost-effectiveness. This paper presents a comprehensive overview of these machines, covering their classification and various applications. The paper starts with discussing the widespread usage of solid-rotor induction machines in numerous industry sectors, including manufacturing, transportation, and renewable energy generation. The ability to operate under harsh environmental conditions and in safety-critical settings has made these machines indispensable in many fields of engineering. Their detailed classification based on different rotor topologies is provided, highlighting the unique design features and performance characteristics of each category. Simple and hybrid configurations and their distinct advantages and limitations in specific applications are included. This paper further explores the essential aspects of multi-physics modeling of solid-rotor induction machines, incorporating electromagnetic, mechanical, and thermal considerations to gain deep insights into the complex interactions between components and to guide the optimization process for enhanced performance and efficiency. This work is intended as a valuable reference for researchers and engineers seeking a comprehensive understanding of solid-rotor induction machines, from their diverse applications to the intricacies of their electromagnetic, thermal, and mechanical modeling. By shedding light on these aspects, this work contributes to the advancement and utilization uptake of these machines in modern industrial settings.