A Banach space is simply a complete normed vector space. This general definition leads to a huge variety in the structure of infinite dimensional Banach spaces. Partly for this reason, when studying Banach spaces we are particularly interested in spaces with additional nice structure, such as having a basis or being reflexive. Given some Banach space, if it does not have a particular property that we are interested in, we want to know if it at least embeds as a subspace of a larger Banach space with that property. A classical example of this is that not every separable Banach space has a basis, but every separable Banach space embeds as a subspace of a Banach space with a basis, namely $C[0,1]$. Likewise, given an operator between two Banach spaces, we want to know when it factors through a Banach space with a particular nice property. The problems of embedding Banach spaces and factoring operators turn out to be directly related. The first half of the talk will be an introduction to the basic theory of Banach spaces, and will cover some classical embedding and factoring theorems. The second half of the talk will cover recent advances in the theory.