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Recent progress in the synthesis of high stability inorganic caesium lead halide perovskite nanocrystals (NCs) with unique optical and electronic properties has led to their increasing use in optoelectronic applications, such as broadband photodetectors. These NCs are of particular interest for the UV range as they have the potential to extend the wavelength range of photodetectors based on traditional materials. The physical properties of these nanostructures, such as the dynamics of charge carriers on different timescales and their effect on the optical recombination of carriers, are crucial for functional applications, but not yet fully understood. This work reports on a slow (> 1 s) reversible quenching of the NC photoluminescence due to a light-induced Stark effect involving defects on the surface of the NCs and the redistribution of photoexcited carriers onto the NC surface. We also demonstrate a defect-assisted high photoconductive gain (10⁶ A/W) in the UV-Visible range for graphene transistors decorated with perovskite NCs, resulting from the charge transfer between the NCs and graphene. This is accompanied by a giant hysteresis of the graphene resistance that is strongly dependent on electrostatic gating and temperature. We summarise the properties of the perovskite/graphene transistor with 3 characteristic response times: optical (∼ 10 s); electrical (∼ 100 s); and magnetic (∼ 500 s). Our data are well described by a phenomenological macroscopic 'two-capacitor' model of the charge transfer from bound states in the NCs into the graphene layer, providing a useful tool for the design of high-photoresponsivity perovskite/graphene transistors. Finally, we investigate the prospects of using scalable additive manufacturing, specifically inkjet printing, of graphene (iGraphene) and other low-dimensional materials for development of fully printed optoelectronic devices compatible with a range of flexible and wearable substrates. We demonstrate a hybrid perovskite/iGraphene photodetector with responsivity ∼15 A/W and use iGraphene as Ohmic contacts to other 2D materials.