broadly speaking, p-adic hodge theory is the study of representations of galois groups of p-adic fields on vector spaces with p-adic coefficients. one can use the theory of $(\varphi,\gamma)$-modules to convert such galois representations into simpler linear algebra, and one can also classify such representations in terms of how arithmetically interesting they are. in my talk, i will discuss extensions of this theory to p-adic families of galois representations. such families arise naturally in the contexts of galois deformation rings and p-adic modular forms.