This project explores a range of computational problems that naturally emerge in the dimension theory of conformal dynamical systems. Conformal fractals are intricate geometric objects generated via iterated schemes of conformal (angle-preserving) transformations, and they have numerous interdisciplinary applications in mathematical physics, computer graphics, and data science. Measuring the size of conformal fractal attractors has been one of the central themes in the evolution of modern dynamical systems. One of the most well-known ways for measuring such complex geometric objects is the concept of Hausdorff dimension, which provides a robust way of determining the roughness of a set, extending the idea of dimension beyond integer values. The Hausdorff dimension of conformal fractals cannot be derived via simple analytic closed formulas, and obtaining effective and rigorous estimates becomes a challenging computational problem. The scope of this project is to introduce new methods from numerical partial differential equations with the scope of developing versatile, rigorous, and efficient methods for computing the Hausdorff dimensions of various conformal attractors. The project's topic is naturally interdisciplinary, encompassing a wide range of problems across Real and Complex Analysis, Dynamical Systems, Numerical Analysis, and Large-Scale Computations. The goal is to derive accurate and rigorous Hausdorff dimension estimates for a broad class of conformal fractals b