Gutzwiller wave function is an important theoretical technique for treating local electron-electron correlations nonperturbatively in condensed matter and materials physics. It is concerned with calculating variationally the ground state wave function by projecting out multi-occupation configurations that are energetically costly. The projection can be carried out analytically in the Gutzwiller approximation that offers an approximate way of calculating expectation values in the Gutzwiller projected wave function. This approach has proven to be very successful in strongly correlated systems such as the high temperature cuprate superconductors, the sodium cobaltates, and the heavy fermion compounds. In recent years, it has become increasingly evident that strongly correlated systems have a strong propensity towards forming inhomogeneous electronic states with spatially periodic superstrutural modulations. A good example is the commonly observed stripes and checkerboard states in high- Tc superconductors under a variety of conditions where superconductivity is weakened. There exists currently a real challenge and demand for new theoretical ideas and approaches that treats strongly correlated inhomogeneous electronic states, which is the subject matter of this thesis. This thesis contains four parts. In the first part of the thesis, the Gutzwiller approach is formulated in the grand canonical ensemble where, for the first time, a spatially (and spin) unrestricted Gutzwiller approximation (SUGA) is developed for studying inhomogeneous (both ordered and disordered) quantum electronic states in strongly correlated electron systems. The second part of the thesis applies the SUGA to the t-J model for doped Mott insulators which led to the discovery of checkerboard-like inhomogeneous electronic states competing with d -wave superconductivity, consistent with experimental observations made on several families of high-Tc superconductors. In the third part of the thesis, new concepts and techniques are developed to study the Mott transition in inhomogeneous electronic superstructures. The latter is termed "SuperMottness" which is shown to be a general framework that unifies the two paradigms in the physics of strong electronic correlation: Mott transition and Wigner crystallization. A cluster Gutzwiller approximation (CGA) approach is developed that treats the local (U) and extended Coulomb interactions (V) on equal footing. It is shown with explicit calculations that the Mott-Wigner metal-insulator transition can take place far away from half-filling. The mechanism by which a superlattice potential enhances the correlation effects and the tendency towards local moment formation is investigated and the results reveal a deeper connection among the strongly correlated inhomogeneous electronic states, the Wigner-Mott physics, and the multiorbital Mott physics that can all be united under the notion of SuperMottness. It is proposed that doping into a superMott insulator can lead to coexistence of local moment and itinerant carriers. The last part of the thesis studies the possible Kondo effect that couples the local moment and the itinerant carriers. In connection to the sodium rich phases of the cobaltates, a new Kondo lattice model is proposed where the itinerant carriers form a Stoner ferromagnet. The competition between the Kondo screening and the Stoner ferromagnetism is investigated when the conduction band is both at and away from half-filling.