The paper introduces Lymph, an open-source MATLAB library for the numerical discretization of coupled multi-physics problems using high-order discontinuous Galerkin (PolyDG) methods on polytopal grids. The library is designed to handle complex geometries, flexible refinement strategies, and heterogeneous physical properties. Key features include geometric flexibility, high-order accuracy, and robustness. The paper covers the installation, input/output data, and code structure of Lymph, providing a user guide that walks through the solution of a Poisson problem. It also demonstrates the library's capabilities through examples such as the Poisson problem, heat equation, and elastodynamics system, showcasing its convergence properties and performance in handling time-dependent problems. The authors highlight the library's potential for various engineering and applied science applications and discuss future developments, including mesh generation algorithms, $hp$-refinement approaches, and extensions to three-dimensional settings.The paper introduces Lymph, an open-source MATLAB library for the numerical discretization of coupled multi-physics problems using high-order discontinuous Galerkin (PolyDG) methods on polytopal grids. The library is designed to handle complex geometries, flexible refinement strategies, and heterogeneous physical properties. Key features include geometric flexibility, high-order accuracy, and robustness. The paper covers the installation, input/output data, and code structure of Lymph, providing a user guide that walks through the solution of a Poisson problem. It also demonstrates the library's capabilities through examples such as the Poisson problem, heat equation, and elastodynamics system, showcasing its convergence properties and performance in handling time-dependent problems. The authors highlight the library's potential for various engineering and applied science applications and discuss future developments, including mesh generation algorithms, $hp$-refinement approaches, and extensions to three-dimensional settings.