Fix a prime number p. The last decade has witnessed multiple foundational advances in algebraic geometry over rings that are p-adically complete (especially in the context of p-adic Hodge theory). This has led to applications in --- and in fact new connections between --- several different areas of mathematics where p-adic completions appear. In this talk, I will survey some of this landscape. Speaker(s): Bhargav Bhatt (University of Michigan)