We present recent results on the Hardy and Bergman spaces of harmonic quasiregular mappings in the unit disk, together with potential extensions to higher-dimensional settings. The talk will cover, in particular, generalizations of classical conjugate function theorems, extremal problems, and criteria for membership in these function spaces. This presentation is based on joint work with Suman Das (Shantou University) and Jie Huang (Huaqiao University).