Martin already mentioned one facet of Stone duality, but I thought I would mention another, which is equally surprising to me. The pro-completion of a category is obtained by formally adjoining cofiltered limits (or equivalently, codirected limits). That is, an object of Pro(C) is a cofiltered diagram in C, with morphisms defined in an appropriate way. Pro(C) can be identified with a full subcategory of $[C,\mathrm{Set}]^{\mathrm{op}}$ (which is the free completion of C under all small limits). Part of Stone duality says that the category Pro(FinSet), the result of formally adjoining cofiltered limits to the category of finite sets, is equivalent to a full subcategory of Top, namely the zero-dimensional compact Hausdorff spaces.