The Fifth International Conference on Finite Fields and Applications Fq5 held at the University of Augsburg, Germany, from August 2-6, 1999 continued a series of biennial international conferences on finite fields. The proceedings document the steadily increasing interest in this topic. Finite fields have an inherently fascinating structure and are important tools in discrete mathematics. Their applications range from combinatorial design theory, finite geometries, and algebraic geometry to coding theory, cryptology, and scientific computing. A particularly fruitful aspect is the interplay between theory and applications which has led to many new perspectives in research on finite fields. This interplay has always been a dominant theme in Fq conferences and was very much in evidence at Fq5. The proceedings reflect this, and offer an up-to-date collection of surveys and original research articles by leading experts in the area.