All invariant contact metric structures on tangent sphere bundles of each compact rank-one symmetric space are obtained explicitly, distinguishing for the orthogonal case those that are K-contact, Sasakian or 3-Sasakian. Only the tangent sphere bundle \(T_{r}{\mathbb {C}} \textbf{P}^{n}\) \((r>0)\) of complex projective spaces admits 3-Sasakian metrics and there exists a unique orthogonal Sasakian-Einstein metric on \(T_{r}{\mathbb {C}} \textbf{P}^{n}.\) Furthermore, there is a unique invariant contact metric that is Einstein, in fact Sasakian-Einstein, on tangent sphere bundles of spheres and real projective spaces. Each invariant contact metric, Sasakian, Sasakian-Einstein or 3-Sasakian structure on the unit tangent sphere of any compact rank-one symmetric space is extended, respectively, to an invariant almost Kähler, Kähler, Kähler Ricci-flat or hyperKähler structure on the punctured tangent bundle.
