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.