p-adic Hodge theory is a branch of mathematics that studies the relationships between p-adic numbers and Hodge theory. It provides tools to understand the behavior of algebraic varieties over p-adic fields, which are extensions of the rational numbers that allow for a different notion of distance. This theory connects various mathematical concepts, such as cohomology and Galois representations, to analyze the properties of p-adic forms. It has applications in number theory and arithmetic geometry, helping mathematicians explore deep connections between different areas of mathematics.