This book presents a comprehensive and geometrical approach to solving partial differential equations (PDEs) on complex curved domains using orthonormal moving frames. Rooted in Élie Cartan’s classical theory but adapted for computational practicality, the framework aligns local basis vectors with the intrinsic geometry and anisotropy of the domain, enabling accurate and efficient discretization without requiring explicit metric tensors or Christoffel symbols. Topics include the construction of moving frames on general manifolds, covariant derivatives via connection 1-forms, and frame-based formulations of gradient, divergence, curl, and Laplacian operators. Extensive MATLAB and C++ implementations (via Nektar++) are provided for benchmark problems in diffusion-reaction systems, shallow water equations, and Maxwell’s equations on complex surfaces such as the sphere, pseudosphere, and atrial tissue. Emphasizing clarity and accessibility, the book blends theory, visualization, and numerical practice, making it an essential resource for graduate students and researchers in scientific computing, applied mathematics, and engineering disciplines dealing with PDEs on non-Euclidean domains.