1.1　Introduction

The electrical resistance of a metallic conductor decreases gradually as temperature drops. Even near absolute zero, metallic conductor still has residual resistance.For a superconductor, the resistance drops abruptly to zero below its critical temperature Tc .

The conventional superconductors are those can be explained by the BCS theory or its derivatives. The superconductors which cannot be explained by the BCS theory are defined as the unconventional superconductors.

The superconductivity was discovered first in mercury by Onnes in 19111，and the perfect diamagnetic phenomena was discovered in 1933. Various superconduct ing theories appeared in succession.Characteristic parameters of superconductors can be well described by phenomenological macroscopic theories.These are the well-known London theory2 and the Ginsburg-Landau theory3.The latter has been extended by Abrikosov4 and Gorkov5 and is therefore often called the GLAG the ory.In 1957，Bardeen, Cooper, and Schrieffer 6 proposed the BCS superconducting microscopic theories.This theory assumed that the charge carriers of the supercon ducting currents are the bound electron pairs（Cooper pairs）and that there exists an energy gap between the normal and superconductive states.

High Tc superconductor（HTSC）is a family of superconducting ceramic mate rials."High"temperature in this context refers to the critical temperature which is higher than the boiling point of liquid nitrogen（77 K or-196℃）.The construction and the operational cost of the cryogenic system for holding the liquid nitrogen are much cheaper than that for liquid helium.The HTSC offers the highest transition temperatures of all superconductors.Thus the HTSC is dominant for commercialapplications since liquid nitrogen is much easier to maintain.For these reasons, the HTSC is particularly attractive for the practical engineering applications.

The London and GLAG theories form a complete set of superconducting macro-scopic theories, and are always used to explain the magnetic properties of type II HTSC. However, these macroscopic theories are approximations of the BCS micro scopic theory.Conventional superconductors can be properly explained by the BCS theory, however, to date, there is no widely accepted theory to explain the super coductivity of HTSC materials as well as other unconventional superconductors.

Superconductors can be classified into type I and type II superconductors. Type I superconductors have a single critical field Hc , above which their superconductiv ity disappears.Type II superconductors have two critical fields, lower critical field Hc1 and upper critical field Hc2 ，between which they allow partial penetration of the magnetic field（mixed state）.

In the past few decades, superconductivity has been described in detail in many works. This chapter only briefly introduces the essential features that directly relate to the theme of this book, high temperature superconducting magnetic levitation, in order to provide the readers with some necessary background knowledge for later chapters.For further details about superconductivity, the related bibliographies are listed in the end of this chapter.