The unstable displacement of one fluid by another in a porous medium occurs frequently in various branches of enhanced oil recovery. It is now well known that when the invading fluid is of lower viscosity than the resident fluid, the displacement front is subject to a Saffman-Taylor instability and is unstable to transverse perturbations. These instabilities can grow, leading to fingering of the invading fluid. Numerical simulation of viscous fingering is challenging. The physics is controlled by a complex interplay of viscous and diffusive forces and it is necessary to ensure physical diffusion dominates numerical diffusion to obtain converged solutions. This typically requires the use of high mesh resolution and high order numerical methods. This is computationally expensive, particularly in 3D. We use IC-FERST, a novel control volume finite element (CVFE) code that uses dynamic mesh adaptivity on unstructured meshes to simulate 2D and 3D viscous fingering with higher accuracy and lower computational cost than conventional methods. We provide evidence that these unstructured mesh simulations in fact yield better results that are less influenced by grid orientation error than their structured counterparts. We also include the effect of capillary pressure and show three examples that are very challenging to simulate using more conventional approaches.