Since the first days of high-T c superconductivity, the materials science and the physics of grain boundaries in superconducting compounds have developed into fascinating fields of research. Unique electronic properties, different from those of the grain boundaries in conventional metallic superconductors, have made grain boundaries formed by high-T c cuprates important tools for basic science. They are moreover a key issue for electronic and large-scale applications of high-T c superconductivity. The aim of this review is to give a summary of this broad and dynamic field. Starting with an introduction to grain boundaries and a discussion of the techniques established to prepare them individually and in a well-defined manner, the authors present their structure and transport properties. These provide the basis for a survey of the theoretical models developed to describe grain-boundary behavior. Following these discussions, the enormous impact of grain boundaries on fundamental studies is reviewed, as well as high-power and electronic device applications. CONTENTS I. Introduction 485 II. Introduction to Grain Boundaries 486 III. Preparation of Single Grain Boundaries 488 A. Bicrystalline junctions 489 B. Biepitaxial junctions 489 C. Step-edge junctions 491 IV. Structural Properties 491 V. Transport Properties of Grain Boundaries 496 A. Current-voltage characteristics 496 B. Critical current density 498 1. Dependence on grain-boundary angle 498 2. Temperature dependence of the critical currents 502 3. Magnetic-field dependence of the critical currents 502 C. Current-phase relation 504 D. Normal-state resistivity 504 E. The I c R n product 505 F. Grain-boundary capacitance 506 G. Microwave properties 507 H. Grain-boundary noise 507 I. Self-generated magnetic flux 509 J. Penetration of magnetic flux into grain boundaries 510 VI. Effects of Doping 510 VII. Grain-Boundary Mechanisms 511 A. Mechanisms based on structural properties 511 B. Mechanisms based on deviations from ideal stoichiometry 512 C. Order-parameter symmetry-based mechanisms 514 D. Interface charging and band bending 515 E. Mechanisms based on direct suppression of the pairing mechanism 516 VIII. Control of Grain Boundaries with Electric Fields or Quasiparticle Injection 517 A. Applied electric fields 517 B. Quasiparticle injection 518 IX. Irradiation of Grain Boundaries 518 A. Irradiation with electrons 518 B. Irradiation with light 518 C. Irradiation with ions 519 X. Bulk Applications 519 A. Powder-in-tube method 520 B. Coated conductors 520 XI. Applications of Grain Boundaries in Thin Films 522 A. SQUIDs 522 B. Radiation detectors and spectrometers 524 C. Three-terminal devices 525 D. Superconducting logic circuits 526 E. Research devices 526 XII. Summary and Outlook 528 Acknowledgments 529 References 529