For oriented 3-manifolds, there are numerous equivalent notions of tautness for co-oriented foliations by surfaces. In this talk, we’ll explore one notion of tautness, the existence of a certain kind of map from the foliated 3-manifold to a surface. This version of tautness was established for sufficiently smooth foliations by Martínez Torres in 2011, and for minimum-regularity foliations by D. Calegari in 2020. We’ll use the definition of branched covering introduced by Fox in 1957 to understand these maps to surfaces. To get our hands dirty, we’ll discuss two opposite ends of the foliation spectrum, foliations with every leaf compact and foliations with every leaf dense. If time permits, we’ll see how Fox’s machinery illuminates a generalization of the behavior of horizontal foliations in Seifert fibered spaces. In particular, every 3-manifold admits a branched circle bundle structure over a branched surface, and taut foliations are exactly the ones that are transverse to such structures. Examples in hyperbolic 3-manifolds will be presented through interactive 3D computer renderings of these branched circle bundle structures.