This book is a substantially rewritten edition of a textbook aimed at students of applied mathematics and engineering. In this version, applications of complex analysis are more concentrated and they cover some important two-dimensional boundary value problems. While the first four chapters are devoted to a detailed exposition of the classical material, culminating in the residue theorem and elementary conformal mapping, the last one deals with applications to fluid mechanics, temperature distribution and electrostatics. The reader is led step by step through the basics of complex function theory. The book contains numerous examples and exercises and selected exercises are provided with solutions. Practical use of the theory is accented; for example the Cauchy theorem forms an optional item. The text is accompanied with many pictures. The book is recommended for technical universities; the exposition is clear and intuitive without going into details of a theoretical nature.