This text forms what is often referred to as "a first course in complex analysis". It is a slight enhancement of lecture notes first presented to undergraduate students in the Mathematics Department of Bedford College, University of London, as part of the Mathematics BSc. degree, and then given for many years in the Mathematics Department of King's College, London. During this time they have been continually revised, reorganized and rewritten. The aim was to provide a rigorous and largely self-contained but extremely gentle introduction to the basics of complex analysis. The audience for the course comprised not only single subject mathematics BSc. and MSci. students but also a number of final year joint honours students as well as postgraduate students who missed out on the subject in their undergraduate programme.