Graphene, a two-dimensional layer of carbon atoms, is a promising building block for a wide range of optoelectronic devices owing to its extraordinary electrical and optical properties, including the ability to absorb ~2% of incident light over a broad wavelength range. While the RC-limited bandwidth of graphene-based photodetectors can be estimated to be as large as 640 GHz, conventional electronic measurement techniques lack for analysing photocurrents at such frequencies. Here we report on time-resolved picosecond photocurrents in freely suspended graphene contacted by metal electrodes. At the graphene-metal interface, we demonstrate that built-in electric fields give rise to a photocurrent with a full-width-half-maximum of ~4 ps and that a photothermoelectric effect generates a current with a decay time of ~130 ps. Furthermore, we show that, in optically pumped graphene, electromagnetic radiation up to 1 THz is generated. Our results may prove essential to build graphene-based ultrafast photodetectors, photovoltaic cells and terahertz sources.