An invisibility cloak should completely hide an object from an observer, ideally across the visible spectrum and for all angles of incidence and polarizations of light, in three dimensions. However, until now, all such devices have been limited to either small bandwidths or have disregarded the phase of the impinging wave or worked only along specific directions. Here, we show that these seemingly fundamental restrictions can be lifted by using cloaks made of fast-light media, termed tachyonic cloaks, where the wave group velocity is larger than the speed of light in vacuum. On the basis of exact analytic calculations and full-wave causal simulations, we demonstrate three-dimensional cloaking that cannot be detected even interferometrically across the entire visible regime. Our results open the road for ultrabroadband invisibility of large objects, with direct implications for stealth and information technology, non-disturbing sensors, near-field scanning optical microscopy imaging, and superluminal propagation.