This paper presents a systematic treatment of the partial differential equations (PDEs) that describe physical systems. The physical world is organized into various systems, such as electromagnetism, perfect fluids, and general relativity, each described by fields on a fixed space-time manifold M. These systems share common features, including the use of fields, partial differential equations, and an initial-value formulation. However, they differ in details such as linearity, constraints, and the nature of their equations. The paper introduces a general framework for first-order, quasilinear PDEs on fields defined on a manifold M. These fields are represented as cross-sections of a fibre bundle, and the PDEs are written on these cross-sections. The framework allows for a broad treatment of physical systems, including the initial-value formulation, constraints, and the geometrical character of physical fields. A key concept is the hyperbolization of PDEs, which involves transforming the system into a symmetric, hyperbolic form. This form allows for the application of a general theorem on the existence and uniqueness of solutions. Constraints are also discussed, which are subsystems of the full PDEs that provide conditions on initial data and lead to differential identities. Constraints are integrable if these identities are truly identities, and complete if they exhaust the full system of equations. The paper also addresses the geometrical character of physical fields, how they transform under diffeomorphisms, and the interactions between different physical fields. It emphasizes that physical fields can interact dynamically (through their derivative terms) or kinematically (through algebraic terms). The paper concludes that the structure of PDEs in physics is best understood through a systematic framework that includes these concepts. The paper also discusses the gauge freedom inherent in the formulation of PDEs and the importance of hyperbolization in ensuring the well-posedness of initial-value problems. The paper concludes that the treatment of PDEs in physics is a blend of analysis, geometry, and physics.This paper presents a systematic treatment of the partial differential equations (PDEs) that describe physical systems. The physical world is organized into various systems, such as electromagnetism, perfect fluids, and general relativity, each described by fields on a fixed space-time manifold M. These systems share common features, including the use of fields, partial differential equations, and an initial-value formulation. However, they differ in details such as linearity, constraints, and the nature of their equations. The paper introduces a general framework for first-order, quasilinear PDEs on fields defined on a manifold M. These fields are represented as cross-sections of a fibre bundle, and the PDEs are written on these cross-sections. The framework allows for a broad treatment of physical systems, including the initial-value formulation, constraints, and the geometrical character of physical fields. A key concept is the hyperbolization of PDEs, which involves transforming the system into a symmetric, hyperbolic form. This form allows for the application of a general theorem on the existence and uniqueness of solutions. Constraints are also discussed, which are subsystems of the full PDEs that provide conditions on initial data and lead to differential identities. Constraints are integrable if these identities are truly identities, and complete if they exhaust the full system of equations. The paper also addresses the geometrical character of physical fields, how they transform under diffeomorphisms, and the interactions between different physical fields. It emphasizes that physical fields can interact dynamically (through their derivative terms) or kinematically (through algebraic terms). The paper concludes that the structure of PDEs in physics is best understood through a systematic framework that includes these concepts. The paper also discusses the gauge freedom inherent in the formulation of PDEs and the importance of hyperbolization in ensuring the well-posedness of initial-value problems. The paper concludes that the treatment of PDEs in physics is a blend of analysis, geometry, and physics.