The ancient view of the night sky as static and unwavering has been rewritten in the past several decades—we now know that “transient” events occur throughout the Universe, producing brief flashes of light that are routinely detected by astronomers. These phenomena are often associated with dramatic cataclysmic events such as stars colliding, exploding, or being torn apart. As such, transients provide a unique testbed for studying physics at the extreme. In recent years, a new window into the dynamic Universe has opened with the groundbreaking detection of gravitational waves. Combining both light and gravitational waves from transient astrophysical sources allows a “multi-messenger” view of the cosmos, akin to sensing the world through both sight and sound. This new lens paves the way to transformational advances that have been reshaping physics and astronomy. In this talk I will give an overview of time-domain and gravitational-wave astrophysics and discuss the power of multi-messenger science. Focusing on binary neutron star mergers as a case study, I will show how multi-messenger inferences have been used to place novel constraints on the (unknown) properties of dense nuclear matter. I will further discuss the future promise of this fast-growing field and its far-reaching implications to other research areas across physics and astronomy.