The book contains three survey articles based on lectures delivered at the University of Tennessee as J. H. Barrett Memorial Lectures. The first part (Partial Differential Equations related to the Gauss-Bonnet-Chern Integrand on 4-manifolds by S.-Y. A.Cheng and P. Yang) is devoted to four dimensional conformal geometry. Conformally invariant curvatures, invariant operators and partial differential equations on 4-dimensional manifolds are studied. In particular, the authors introduce the Weyl curvature tensor and Q-curvature and they study conformal compactification of a complete non-compact locally conformally flat four-manifold with integrable Q. It includes also a study of properties of the second elementary symmetric function σ2(A) , where A is the conformal Ricci tensor. The second part by K. Grove (Geometry of, and via, Symmetries) is devoted to a study of properties of the isometry group of Riemannian manifolds and its geometry and topology. Several known examples are presented, the Alexandrov geometry of orbit spaces is described and studied using symmetries. In the last section open problems and conjectures are stated. The third part (The geometry of Lagrangian immersions into symplectic manifolds by J. G. Wolfson) is devoted to a study of Lagrangian immersions and their invariants and to the problem of minimizing volume among Lagrangian cycles inside a Lagrangian homology class.