This thesis is an overview of the geometry of nearly Kähler six-manifolds. A nearly Kähler structure on a manifold M is a special kind of non-integrable Hermitian structure (g,J) quite different from a Kähler structure. Dimension six is particularly interesting due to connections with the exceptional holonomy group G_2 in dimension seven. Moreover, the classification in arbitrary dimension is reducible to that in dimension six, where there are only four known examples, homogeneous structures on S^6, S^3 X S^3, CP^3 and the variety of flags in C^3.

With this background, we review the geometry of six dimensional nearly Kähler manifolds and explore examples. We prove that every nearly Kähler six-manifold (M,g,J) is Einstein, establish the connection with G_2 geometry and prove an important uniqueness theorem for complete nearly Kähler six-manifolds.

This final result is particularly important for the consideration of group actions: if (M,g,J) is a complete nearly Kähler six-manifold not isometric to a round sphere and G is a group of isometries of (M,g), then G preserves J also. Nearly Kähler six-manifolds with a large degree of symmetry are studied in some depth in this thesis. We review the works of Butruille and Podestà-Spiro where a group of isometries acts transitively and with codimension one, respectively. The former problem is solved completely, producing just the four examples alluded to above. In the latter situation, M must be one of S^6, S^3 X S^3 or projective 3-space, and SU(3) X SU(3) is the only interesting group that can act. The classification of nearly Kähler structures on M is reduced to a system of non-linear ODE which, following Podestà-Spiro, we make the first steps in analysing. The work falls short of a classification of complete cohomogeneity one nearly Kähler six-manifolds and we conclude with a summary of the work remaining to be done in this direction.