  This paper is written as a brief introduction for beginning graduate
students. The picture of electron waves moving in a cristalline potential and
interacting weakly with each other and with cristalline vibrations suffices to
explain the properties of technologically important materials such as
semiconductors and also simple metals that become superconductors. In magnetic
materials, the relevant picture is that of electrons that are completely
localized, spin being left as the only relevant degree of freedom. A number of
recently discovered materials with unusual properties do not fit in any of
these two limiting cases. These challenging materials are generally very
anisotropic, either quasi one-dimensional or quasi two-dimensional, and in
addition their electrons interact strongly but not enough to be completely
localized. High temperature superconductors and certain organic conductors fall
in the latter category. This paper discusses how the effect of low dimension
leads to new paradigms in the one-dimensional case (Luttinger liquids,
spin-charge separation), and indicates some of the attempts that are being
undertaken to develop, concurrently, new methodology and new concepts for the
quasi-two-dimensional case, especially relevant to high-temperature
superconductors.
