The growing importance of applications based on machine learning is driving the need to develop dedicated, energy-efficient electronic hardware. Compared with von Neumann architectures, which have separate processing and storage units, brain-inspired in-memory computing uses the same basic device structure for logic operations and data storage, thus promising to reduce the energy cost of data-centred computing substantially. Although there is ample research focused on exploring new device architectures, the engineering of material platforms suitable for such device designs remains a challenge. Two-dimensional materials such as semiconducting molybdenum disulphide, MoS2, could be promising candidates for such platforms thanks to their exceptional electrical and mechanical properties. Here we report our exploration of large-area MoS2 as an active channel material for developing logic-in-memory devices and circuits based on floating-gate field-effect transistors (FGFETs). The conductance of our FGFETs can be precisely and continuously tuned, allowing us to use them as building blocks for reconfigurable logic circuits in which logic operations can be directly performed using the memory elements. After demonstrating a programmable NOR gate, we show that this design can be simply extended to implement more complex programmable logic and a functionally complete set of operations. Our findings highlight the potential of atomically thin semiconductors for the development of next-generation low-power electronics.