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A smooth foliation is Riemannian when its leaves are locally equidistant. Theclosures of the leaves of a Riemannian foliation on a simply connectedmanifold, or more generally of a Killing foliation, are described by flows oftransverse Killing vector fields. This offers significant technical advantagesin the study of this class of foliations, which nonetheless includes otherimportant classes, such as those given by the orbits of isometric Lie groupactions. Aiming at a broad audience, in this survey we introduce Killingfoliations from the very basics, starting with a brief revision of the mainobjects appearing in this theory, such as pseudogroups, sheaves, holonomy andbasic cohomology. We then review Molino's structural theory for Riemannianfoliations and present its transverse counterpart in the theory of completepseudogroups of isometries, emphasizing the connections between these topics.We also survey some classical results and recent developments in the theory ofKilling foliations. Finally, we review some topics in the theory of singularRiemannian foliations and discuss singular Killing foliations.

Leaf closures of Riemannian foliations: a survey on topological and geometric aspects of Killing foliations