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The present notes contain the foundations of algebraic K-theory to- gether with a series of explicit computations of the K-groups of fields and some classical varieties. Our goal is to provide an introduction to a more advanced reading, as well as to convince the reader that such a study may be useful and interesting. The exposition is by no means complete nor self-contained. We hope nevertheless, that the covered part of the theory is sufficient for effective computations in algebraic geometry.