Gravity is successfully described by Einstein’s theory of general relativity (GR), governing the structure of our entire universe. Yet it remains the least understood of all forces in nature, resisting unification with quantum physics. One of the most fundamental predictions of GR are black holes (BHs). Their defining feature is the event horizon, the surface that light cannot escape and where time and space exchange their nature. However, while there are many convincing BH candidates in the universe, there is no experimental proof for the existence of an event horizon yet. So, does GR really hold in its most extreme limit? Do BHs exist or are alternatives needed? Here we propose to build a Black Hole Camera that for the first time will take an actual picture of a BH and image the shadow of its event horizon. We will do this by providing the equipment and software needed to turn a network of existing mm-wave radio telescopes into a global interferometer. This virtual telescope, when supplemented with the new Atacama Large Millimetre Array (ALMA), has the power to finally resolve the supermassive BH in the centre of our Milky Way – the best-measured BH candidate we know of. In order to compare the image with the theoretical predictions we will need to perform numerical modelling and ray tracing in GR and alternative theories. In addition, we will need to determine accurately the two basic parameters of the BH: its mass and spin. This will become possible by precisely measuring orbits of stars with optical interferometry on ESO’s VLTI. Moreover, our equipment at ALMA will allow for the first detection of pulsars around the BH. Already a single pulsar will independently determine the BH’s mass to one part in a million and its spin to a few per cent. This unique combination will not only produce the first-ever image of a BH, but also turn our Galactic Centre into a fundamental-physics laboratory to measure the fabric of space and time with unprecedented precision.