These notes record, in a slightly expanded way, the lectures given by the first two authors at the College on Moduli Spaces of Vector Bundles that took place at CIMAT in Guanajuato, Mexico, from November 27th to December 8th, 2006. The college, together with the ensuing conference on the same topic, was held in occasion of Peter Newstead's 65th anniversary. It has been a great pleasure and a privilege to contribute to celebrate Peter's outstanding achievements in algebraic geometry and his lifelong dedication to the progress of mathematical knowledge. We warmly thank the organizers of the college and conference for inviting us, thus allowing us to participate in Peter's celebration.

The main emphasis in these notes is on the Fourier-Mukai transforms as equivalences of derived categories of coherent sheaves on algebraic varieties. For this reason, the first Section is devoted to a basic (but we hope, understandable) introduction to derived categories. In the second Section we develop the basic theory of Fourier-Mukai transforms.

Another aim of our lectures was to outline the relations between Fourier-Mukai and Nahm transforms. This is the topic of Section 3. Finally, Section 4 is devoted to the application of the theory of Fourier-Mukai transforms to the study of coherent systems.

This is a review paper. Most of the material is taken from [BBH08] and [HT08], although the presentation is different in some places.