This book provides a quick access to computational tools for algebraic geometry, the mathematical discipline which handles solution sets of polynomial equations. Originating from a number of intense one week schools taught by the authors, the text is designed so as to provide a step by step introduction which enables the reader to get started with his own computational experiments right away. The authors present the basic concepts and ideas in a compact way, omitting proofs and detours, and they give references for further reading on some of the more advanced topics. In examples and exercises, the main emphasis is on explicit computations using the computer algebra system SINGULAR. The book addresses both, students and researchers. It may serve as a basis for self-study, guiding the reader from his first steps into computing to writing his own procedures and libraries.