Eccentric binary systems are ubiquitous. They appear in many astrophysical contexts, ranging from migrating giant planets to coalescing neutron star (NS) binaries. In a highly eccentric binary, the separation between the two bodies can vary greatly over the course of an orbit. Yet any binary with a small minimum separation (even if it lasts a scant fraction of an orbit) can be shaped dramatically by the tidal distortion and heating of one or both bodies. Tides are commonly treated in a parameterized way for convenience. In truth, a body's efficiency at dissipating energy depends strongly on its structure and orbit. This dissertation explores how the nuances of tidal physics can dramatically alter the expected orbital evolution of many types of binary systems. In Chapter 2, I demonstrate that a white dwarf (WD) on an eccentric orbit around a massive black hole (MBH) can experience significant tidal heating, long before it is torn apart by the tidal force of the BH. The WD-MBH pairing is especially interesting because the inevitable outcome --- the tidal disruption of the WD --- will be visible at cosmological distances, but only if the BH mass is less than 100,000 solar masses. These events could constrain the mass function of intermediate mass to massive BHs. In Chapter 3, I develop a simple model that captures the coupled evolution of a binary orbit and tidally excited oscillations in one (or both) of its components. I derive the conditions under which the oscillations can grow chaotically over successive orbits, and explore how the damping of these oscillations affects long-term orbital evolution. Chapter 4 applies the model from Chapter 3 to the high-eccentricity migration of giant planets. I study how, when a giant planet is excited onto a highly eccentric orbit, chaotic tides can rapidly drain energy from the orbit and shrink the semi-major axis. I demonstrate that chaotic tides can resolve many of the difficulties facing the high-eccentricity migration theory of hot Jupiter formation. In Chapter 5, I analyze the role of tides in the coalescence of an eccentric neutron star binary. Measuring the tidal response of neutron stars would help to unveil their equation of state. Lastly, in Chapters 6 and 7, I develop a treatment for tidal dissipation via turbulent viscosity in a star with a convective envelope in an eccentric orbit and apply this formalism to pre-common envelope binaries with massive stars. As the massive star evolves off of the main sequence and develops a deep convective envelope, viscous dissipation in the star circularizes and shrinks the binary. However, I find that, in many cases, tidal dissipation cannot circularize the orbit before the binary reaches Roche-lobe overflow.