In this chapter, we study the relationship between a complete nonarchimedean field and its finite extensions; this relationship involves the residue fields, value groups, and Galois groups of the fields in question. We distinguish some important types of extensions, the unramified and tamely ramified extensions. We also briefly discuss the special case of discretely valued fields with perfect residue field, in which one can say much more. We introduce the standard ramification filtrations on the Galois groups of extensions of local fields; these will not reappear again until Part IV, at which point they will relate to the study of convergence of solutions of p-adic differential equations made in Part III.