Abstract: The accuracy of atomic clocks has improved a thousandfold over the last 15 years, driven by improvements in ultrastable lasers, quantum control, and our understanding of atomic interactions. The latest generation of optical lattice clocks are accurate enough to measure general relativity's gravitational redshift at the centimeter scale, to test physics beyond the Standard Model, and to reveal new emergent properties of quantum materials. In this talk, I will discuss the principles behind state-of-the-art optical lattice clocks and describe a clock that reaches a record accuracy of 1 mHz on an optical transition at 429 THz, a fractional frequency accuracy of 2 parts in 10^18. I will also discuss a next-generation clock that probes degenerate fermions in a three-dimensional optical lattice. By controlling atomic interactions and light shifts, this apparatus achieved a record atom-light coherence time of six seconds, reaching sub-100 mHz spectroscopic features. Finally, if time allows, I will discuss prospects for improved accuracy in a cryogenic strontium clock and for tests of fundamental physics with clocks.