This volume contains reprints of 30 scientific articles which provide a record of some of the main advances in understanding which have lead us from Edwin Hall's discovery of the effect which bears his name to our current understanding of the quantum Hall efFect. In the process of deciding which articles were to be included in the collection and which were to be left out I became aware of the gaps in my knowledge of the relevant history, both very recent and more distant. The final choices are, I have no doubt, imperfect. I know that they reflect to too large a degree the aspects of, the subject on which I have 'worked myself, so that the collection represents mil perspective on the quantum Hall effect. It is my hope ,however, that the reader will find that these articles taken together have a rather interesting story to tell: a story which reminds us of the unpredictable consequences of scientific enquiry and reflects the impressive achievements of condensed matter physics in this century. The reprints are accompanied by an Introduction which can stand on its own as a brief explanation of the quantum Hall efFect or serve as an primer for the reprinted articles. The point of view taken in the introduction is again my own and is not the only useful one.

A. Overview.- B. The Basics.- C. The Integer Quantum Hall Effect.- D. The Quantum Hall Effect in a Periodic Potential.- E. The Fractional Quantum Hall Effect.- References:.- Reprinted Articles.- The Discovery of the Hall Effect: Ewdin Hall's Hiterto Unpublished Account, in "The Hall effect and its applications" (1980).- Diamagnetismus der Metalle, Zeitscrift fur Physik (1930).- Der Diamagnetismus von frein Elektronen, Zeitscrift fur Physik (1931).- The Dependence of the Susceptibility of Diamagnetic Metals upon the Field, Kon. Akad. van Wefenschappen (1930).- Magnetische Widerstandsvergroserung in Einkristallen von Wismut bei Tiefen Temperaturen, Kon. Akad. van Wefenschappen (1930).- Magnetoresistance, in "The Fermi Surface" (1960).- Magneto-oscillatory Conductance in Silicon Surfaces, Phys. Rev. Lett. (1966).- Theory of Hall Effect in a two-Dimensional Electron System, J. Phys. Soc. Jpn. (1975).- Quantum Galvanomagnetic Properties of n-Type Inversion Layers on Si(100) MOSFET, Surf. Sci. (1976).- The Discovery of Quantum Hall Effect, Metrologia (1986).- New method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance, Phys. Rev. Lett. (1980).- Quantized Hall Resistance and the Measurement of the Fine-Structure Constant, Phys. Rev. (1981).- Quantized Hall Conductivity in two Dimensions, Phys. Rev. (1981).- Quantized Hall Conductance, Current Carrying Edge States and the Existence of Extended States in a two Dimensional Disordered Potential, Phys. Rev. (1982).- Theory of Quantized Hall Conductivity in two Dimensions, J. Phys. C (1982).- Localization and Scaling in the Quantum Hall Regime, Phys. Rev. (1985).- Quantization of Coupled Orbits in Metals, Proc. Roy. Soc. (1962).- Energy Levels and Wave Functions of BlochElectrons in Rational and Irrational Magnetic Fields, Phys. Rev. (1976).- Quantized Hall Effect in Two Dimensional Periodic Potentials, Phys. Rep. (1984).- Quantised Hall Effect in a Two Dimensional Periodic Potential, J. Phys. C (1982).- Two Dimensional Magnetotransport in the Extreme Quantum Limit, Phys. Rev. Lett. (1982).- Odd and Even Fractionally Quantized States in GaAs-GaAlAs Heterojunctions, Surf. Sci. (1986).- Magnetic Field Dependence of Activation Energies in the Fractional Quantum Hall Effect, Phys. Rev. Lett. (1985).- Second activation Energy in the Fractional Quantum Hall Effect, J. Phys. Soc. Jpn. (1987).- Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations, Phys. Rev. Lett. (1983).- Fractional Quantization of the Hall Effect: A Hierarchy of Incompressible Quantum Fluid States, Phys. Rev. Lett. (1983).- Ground State of Two Dimensional Electrons in Strong Magnetic Fields and $$\frac{1}{3}$$ Quantized Hall Effect, Phys. Rev. Lett. (1983).- Statistics of Quasiparticles and the Hierarchy of Fractional Quantized Hall States, Phys. Rev. Lett. (1984).- Magneto-roton Theory of Collettive Excitations in the Fractional Quantum Hall Effect, Phys. Rev. (1986).- Off Diagonal Long-Range Order, Oblique Confinement, and the Fractional Quantum Hall Effect, Phys. Rev. Lett. (1987).