The Lagrangian submanifolds (LS) in Kähler manifolds and in the nearly Kähler six-sphere are studied from Riemannian point of view. The basic properties of LS are reviewed and the Riemannian obstructions to Lagrangian isometric immersions are searched. Optimal inequalities between scalar curvature, Ricci curvature, shape operator and mean curvature are obtained. The LS with parallel mean curvature vector are considered as natural generalizations of minimal submanifolds. The following examples are given: totally geodesic LS, ideal Lagrangian isometric immersions, H-umbilical LS. The index and stability of LS are studied and criteria of stability are established. The Maslov class of LS is defined and some of its properties are obtained for the Einstein-Kähler manifolds. The case of the nearly Kähler six-sphere is analyzed in detail and a classification of its LS is given.