Registration, which aims to find an optimal one-to-one correspondence between different data, is an important problem in various fields. This problem is especially challenging when large deformations occur. In this paper, we present a novel algorithm to obtain diffeomorphic image or surface registrations with large deformations via quasi-conformal maps. The basic idea is to minimize an energy functional involving a Beltrami coefficient term, which measures the distortion of the quasi-conformal map. The Beltrami coefficient effectively controls the bijectivity and smoothness of the registration. Using the proposed algorithm, landmark-matching diffeomorphic (1-1 and onto) registrations between images or surfaces can be effectively obtained, even with a large deformation or large number of feature landmark constraints. The proposed algorithm can also be extended to a hybrid registration model, called Q-Fibra, which combines landmark and intensity (such as image intensity or surface curvature) information to obtain a more accurate registration. Experiments have been carried out on both synthetic and real data. Results demonstrate the efficacy of the proposed algorithm to obtain diffeomorphic registrations between images or surfaces.