  Einstein's General Theory of Relativity (GR) successfully describes gravity.
The most fundamental predictions of GR are black holes (BHs), but in spite of
many convincing BH candidates in the Universe, there is no conclusive
experimental proof of their existence using astronomical observations in the
electromagnetic spectrum. Are BHs real astrophysical objects? Does GR hold in
its most extreme limit or are alternatives needed? The prime target to address
these fundamental questions is in the center of our own Galaxy, which hosts the
closest and best-constrained supermassive BH candidate in the Universe,
Sagittarius A* (Sgr A*). Three different types of experiments hold the promise
to test GR in a strong-field regime using observations of Sgr A* with
new-generation instruments. The first experiment aims to image the relativistic
plasma emission which surrounds the event horizon and forms a "shadow" cast
against the background, whose predicted size (~50 microarcseconds) can now be
resolved by upcoming VLBI experiments at mm-waves such as the Event Horizon
Telescope (EHT). The second experiment aims to monitor stars orbiting Sgr A*
with the upcoming near-infrared interferometer GRAVITY at the Very Large
Telescope (VLT). The third experiment aims to time a radio pulsar in tight
orbit about Sgr A* using radio telescopes (including the Atacama Large
Millimeter Array or ALMA). The BlackHoleCam project exploits the synergy
between these three different techniques and aims to measure the main BH
parameters with sufficient precision to provide fundamental tests of GR and
probe the spacetime around a BH in any metric theory of gravity. Here, we
review our current knowledge of the physical properties of Sgr A* as well as
the current status of such experimental efforts towards imaging the event
horizon, measuring stellar orbits, and timing pulsars around Sgr A*.
