There is a long history of counting permutations according to statistics such as descents and peaks, with connections to geometry and algebra. A descent in a permutation w is a position i such that w(i)>w(i+1), while a peak is a position i such that w(i-1)<w(i)>w(i+1). The number of permutations with a fixed descent set is well-known, and not too long ago Billey, Burdzy, and Sagan explored the analogous question for peak sets. In recent work with Rob Davis, Sarah Nelson, and Bridget Tenner, we study what happens when we record not "i" but rather "w(i)" for each peak. (A similar variation on descents can be found in work of Ehrenborg and Steingrimsson.) We call these values "pinnacles" and ask the basic question: How many permutations have a given set of pinnacles?