The chapter introduces a systematic treatment of the partial differential equations (PDEs) that describe various physical systems, such as electromagnetism, perfect fluids, Klein-Gordon fields, elastic media, and gravitation. These systems share common features, such as the use of fields on a fixed spacetime manifold \( M \), geometrical interpretations, initial-value formulations, and constraints. However, they also differ in details, such as linearity, the presence or absence of constraints, and the methods of derivation from Lagrangians. The author proposes a general framework for PDEs on fields defined on a fiber bundle over \( M \), where the fields are interpreted as cross-sections of the bundle. This framework allows for a broad range of physical systems but is not suitable for explicit calculations. The chapter discusses several key aspects of these PDE systems: 1. **Hyperbolizations**: A hyperbolization of a PDE system is a smooth field on the bundle such that the system can be cast into a symmetric, hyperbolic form. This form allows for a well-posed initial-value formulation, where the solution is uniquely determined by initial data. The existence and uniqueness theorem guarantees that, given initial data, there is a unique solution in a suitable neighborhood. 2. **Constraints**: Constraints are tensor fields that play a dual role. They provide differential conditions that must be satisfied by initial data and lead to differential identities involving the PDE system. The constraints are complete if the number of equations matches the number of unknowns, ensuring that all relevant conditions on the first derivative of the cross-section are included. 3. **Geometrical Character of Fields**: The transformation properties of the fields under diffeomorphisms on \( M \) are discussed, emphasizing the importance of diffeomorphism invariance in the initial-value formulation. 4. **Interactions Between Fields**: The chapter explores how different physical fields can interact dynamically (through derivative terms) and kinematically (through algebraic terms). It highlights that fields are part of the same physical system if their derivative terms cannot be separated, and one field serves as a background for another if it appears algebraically in the derivative terms of the latter. The chapter concludes with a discussion of specific examples, including electromagnetism, perfect fluids, and general relativity, to illustrate the concepts introduced. It also addresses the challenges and limitations of the approach, such as the lack of a practical procedure for finding hyperbolizations in general cases and the behavior of constraints in certain systems like dust and Einstein's equations.The chapter introduces a systematic treatment of the partial differential equations (PDEs) that describe various physical systems, such as electromagnetism, perfect fluids, Klein-Gordon fields, elastic media, and gravitation. These systems share common features, such as the use of fields on a fixed spacetime manifold \( M \), geometrical interpretations, initial-value formulations, and constraints. However, they also differ in details, such as linearity, the presence or absence of constraints, and the methods of derivation from Lagrangians. The author proposes a general framework for PDEs on fields defined on a fiber bundle over \( M \), where the fields are interpreted as cross-sections of the bundle. This framework allows for a broad range of physical systems but is not suitable for explicit calculations. The chapter discusses several key aspects of these PDE systems: 1. **Hyperbolizations**: A hyperbolization of a PDE system is a smooth field on the bundle such that the system can be cast into a symmetric, hyperbolic form. This form allows for a well-posed initial-value formulation, where the solution is uniquely determined by initial data. The existence and uniqueness theorem guarantees that, given initial data, there is a unique solution in a suitable neighborhood. 2. **Constraints**: Constraints are tensor fields that play a dual role. They provide differential conditions that must be satisfied by initial data and lead to differential identities involving the PDE system. The constraints are complete if the number of equations matches the number of unknowns, ensuring that all relevant conditions on the first derivative of the cross-section are included. 3. **Geometrical Character of Fields**: The transformation properties of the fields under diffeomorphisms on \( M \) are discussed, emphasizing the importance of diffeomorphism invariance in the initial-value formulation. 4. **Interactions Between Fields**: The chapter explores how different physical fields can interact dynamically (through derivative terms) and kinematically (through algebraic terms). It highlights that fields are part of the same physical system if their derivative terms cannot be separated, and one field serves as a background for another if it appears algebraically in the derivative terms of the latter. The chapter concludes with a discussion of specific examples, including electromagnetism, perfect fluids, and general relativity, to illustrate the concepts introduced. It also addresses the challenges and limitations of the approach, such as the lack of a practical procedure for finding hyperbolizations in general cases and the behavior of constraints in certain systems like dust and Einstein's equations.