The magnetoresistance (MR), that means the resistance change induced by an external magnetic field, is phenomena observed, more or less, in all metals and semiconductors . The particular MR phenomena to be described here are the gigantic decrease of resistance by application of a magnetic field that is observed for the transition metal oxides and arises from the spin-dependent scattering process of the conduction electrons. Importantly, the local spins and conduction electrons are both of d-electron character. In Figure 1 is shown a typical example of the MR feature observed for a single crystal of Lai-^Ca^MnOs (x = 0.33) with perovskite-like structure. Detailed description and account for this “colossal” MR feature will given in the following sections. Here, the reader may notice that an external magnetic fields cause the gigantic decrease of the resistivity around the Curie temperature of this compound, below which the resistivity also shows a steep decrease with decreasing temperature in zero field. Such gigantic negative MR, now termed “colossal magnetoresistance” (CMR), in the perovskite manganites near the Curie temperature (7c) seems to have already been known at the very early stage of the study on transition metal oxides. For example, the paper in 1969 by Searle and Wang  reported thoroughly the magnetic field dependence of the resistivity for a flux-grown crystal of Lai_xPbxMn03, in particular the large MR near Tc, as well as the phenomenological analysis. Soon after, Kubo and Ohata  have given a theoretical account for this phenomenon using the so-called double-exchange Hamiltonian (the essentially s-d model with the ferromagnetic on-site exchange interaction), that includes the essential ingredient of the double exchange mechanism elaborated by Zener , Anderson and Hasegawa , and de Gennes . Interest in the MR of those manganites has revived more lately since the rediscovery of the CMR or even more astonishing magnetic field induced insulator-metal [7-11] and lattice-structural  transitions. On one hand, the highly sensitive and electrically-readable magnetic-field sensors have recently
Figure 1: Temperature profile o f resistivity o f a L a \-xCaxMnO^ (x = 0.3) single crystal under several magnetic fields applied parallel to the electric current.