Metric theory offers powerful tools for analysing and processing shapes on curved manifolds, with Riemannian metrics being the most commonly used due to their simplicity and success in many applications. However, Riemannian metrics are limited by their symmetric nature, where lengths of paths are independent of traversal direction. Finsler metrics, a generalisation including asymmetric distances, provide broader tools but are rarely applied in practice, perhaps due to their theoretical complexity. This talk describes Finsler metrics, showcasing their potential through two applications. First, we revisit anisotropy in Laplace-Beltrami operators (LBO) on Riemannian manifolds. By exploring a Finsler perspective, we design a new Riemannian LBO with Finsler-movitaved anisotropy, and apply it for shape matching. Second, we revisit heuristic deformation strategies in image convolution, proposing a unifying theory where kernel positions are samples of unit balls from implicit metrics. Introducing metric convolutions, which involve sampling of unit balls of explicit metrics and are compatible with neural networks, we achieve competitive results in denoising and classification tasks.