Transition metal oxides whose lattice structure has edge-sharing network of octahedra constitute a diverse group of intriguing materials besides compounds with corner-sharing octahedra such as perovskites. We present a theoretical investigation of the interplay among spin, orbital and lattice degrees of freedom in these materials. We focus on t2g electron systems where a keen competition among those degrees of freedom is expected to emerge under a relatively weak Jahn-Teller coupling. We study the interplay between spin and orbital degrees of freedom in vanadium spinels and titanium pyroxenes. We clarify the important role of the strong anisotropy in the orbital interactions due to the edge-sharing geometry. We also discuss the interplay between spin and lattice in chromium spinels focusing on the magnetization process under the external magnetic field. §1. IntroductionTransition metal oxides, in many cases, consist of the basic unit of octahedron where a metal is surrounded by six oxygens. One of the famous families is the perovskite such as high-T c cuprates and CMR manganites, in which octahedra form two-dimensional (2D) or three-dimensional (3D) network by sharing oxygens at their corner. Another typical geometry is the edge-sharing network of octahedra. An old but still intriguing example is the spinel in which octahedra form 3D edge-sharing network. A 2D example is found in the sodium cobaltite which has a triangular lattice of Co cations. There, a large thermoelectric effect or superconductivity is recently attracting much interest.In this paper, we present our recent theoretical efforts to understand remarkable properties in several edge-sharing materials with focusing on a keen competition among spin, orbital and lattice degrees of freedom. In the octahedral coordinate, the fivefold energy levels of d electrons of transition metals split into lower threefold t 2g levels and higher twofold e g levels. Here, we consider the systems in which electrons in the t 2g levels play a central role. In the t 2g electron systems, it is known that the Jahn-Teller interaction is rather weak compared to the e g systems.