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The main purpose of these lecture notes is to provide a concise introductionto Lie groups, Lie algebras, and isometric and adjoint actions, aiming mostlyat advanced undergraduate and graduate students. In addition, the connectionbetween such classic theories and the research area of the first author isexplored. Namely, generalizations to isoparametric submanifolds, polar actionsand singular Riemannian foliations with sections (s.r.f.s.) are mentioned. Thefirst chapters cover basic concepts, giving results on adjoint representation,closed subgroups, bi-invariant metrics, Killing forms and splitting in simpleideals. In the following chapters, proper and isometric actions are recalledtogether with adjoint action and foliations, mostly concerning the Weyl group,normal slices and Dynkin diagrams. A special focus is given to maximal tori androots of compact Lie groups, exploring its connection with isoparametricsubmanifolds and polar actions. Furthermore, in the last chapter, a survey onrecent research results on s.r.f.s. is given.  In this revised version, more details about fiber bundles, proper andisometric actions are explored, and further exercises and examples were added.It also features new sections with examples of singular Riemannian foliationsconstructed with surgery and suspension of homomorphisms. This is still apreliminary version and we expect to improve it in the future. We would begrateful for any kind of suggestions.

Introduction to Lie groups, isometric and adjoint actions and some generalizations