In this overview, the recent theoretical land some experimental) works on a variety of physical properties that arise from correlations among electrons interacting via Coulomb interactions in three and two - dimensional systems will be discussed. The subject matter has a long 70 year history which we meander through in this brief presentation, even after exclusion of several important aspects of the problem. The astrophysical and nuclear physics aspects of these problems will not be discussed. The main focus will be concerning issues of condensed matter physics. Also not discussed in this presentation is the important works on the electron correlations based on the Slater-Hubbard model (with a relatively shorter 40-year history) which is central in the recent discussions of strongly coupled systems such as high temperature superconductors and magnetic materials. Excluded also from discussion here is the effects of disorder, which is a separate topic in itself. Included in this talk will be spin-polarization as well as finite temperature effects, in both bulk and semi-infinite situations, and electron-hole plasmas. Interesting physical situations of two-dimensionality occuring in Mosfets, semiconductor heterojunctions, and electrons on cylindrical surfaces as in carbon nanotubules, will be briefly touched upon as they possess rich consequences of correlations. The effects of quantizing magnetic fields and the relativistic situations will only be mentioned in passing. Theoretical techniques used fall basically into five categories in my classification: (1) wave function methods - variational and nonvariational, (2) phenomenological/intuitive methods subsumed by diagrammatic techniques, (a) collective excitation theory of Bohm-Pines leading to Boson formulation, (b) dielectric formulation of Singwi and coworkers, (c) Fermi liquid theory of Landau, culminating in diagrammatic perturbation theory of Gell-Mann and Brueckner which in various forms contains all these and had important off-shoots, (3) Quantum Monte Carlo methods, (4) the method of Green functions, and finally, (5) density-functional method of Kohn and coworkers. Each of these had different insights to offer which we will spell out. Very recent work on the single particle Green function will also be discussed because of its implications to several physical properties of the system. A brief discussion of pair correlations and response functions is given as this provides information on collective properties such as plasma and spin wave oscillations, etc.
