The concentration of measure phenomenon is a mathematical equivalent to the saying "what you see is what you get", as long as you need glasses and are not wearing any, or are looking from far away. A consequence of this phenomenon in a Gaussian measure setting is that Lipschitz functions are "almost constant". Following a version of Vitali Milman's proof, we apply this consequence to the norms on Banach spaces to prove that every convex, compact, and symmetric body in n-dimensional Euclidean space has a relatively high dimensional slice that is "almost ellipsoidal". This is the content of Dvoretzky's theorem, which has numerous applications in functional analysis, including its important applications to the local theory of Banach spaces. Time permitting, we will discuss some of these applications, as well as an extension of a weak version of Dvoretzky's theorem to operator spaces, due to Gilles Pisier.