These notes had their origin in a postgraduate lecture series I gave at the Eid genossiche Technische Hochschule (ETH) in Zurich in the Spring of 2000. I am very grateful to my hosts, the Forschungsinstitut fUr Mathematik at ETH, for providing the ideal opportunity to develop and present this material in what I hope is a reasonably coherent manner, and also for encouraging and assisting me to record the proceedings in these lecture notes. The subject of the lecture series was counting (of combinatorial structures) and related topics, viewed from a computational perspective. As we shall see,...