Rapid digital technology advancement has resulted in a tremendous increase in computing tasks imposing stringent energy efficiency and area efficiency requirements on next-generation computing. To meet the growing data-driven demand, in-memory computing and transistor-based computing have emerged as potent technologies for the implementation of matrix and logic computing. However, to fulfil the future computing requirements new materials are urgently needed to complement the existing Si complementary metal-oxide-semiconductor technology and new technologies must be developed to enable further diversification of electronics and their applications. The abundance and rich variety of electronic properties of two-dimensional materials have endowed them with the potential to enhance computing energy efficiency while enabling continued device downscaling to a feature size below 5 nm. In this Review, from the perspective of matrix and logic computing, we discuss the opportunities, progress and challenges of integrating two-dimensional materials with in-memory computing and transistor-based computing technologies.