The consecutive pattern poset is the infinite partially ordered set of all permutations, where $\sigma\le\tau$ if $\tau$ has a subsequence of adjacent entries in the same relative order as the entries of $\sigma$. We study the structure of the intervals in this poset from topological, poset-theoretic, and enumerative perspectives. Among other results, we classify the intervals of the following types: disconnected; shellable; rank-unimodal; strongly Sperner. This is joint work with Sergi Elizalde.