This book solves a problem that has been open for over 20 years - the complete classification of structurally stable quadratic vector fields modulo limit cycles. The 1950s saw the first real impetus given to the development of the qualitative theory of quadratic vector fields, although prior and ongoing interest in the topic can be shown by the more than 800 papers that have been published on the subject. One of the problems in the qualitative theory of quadratic vector fields is the classification of all structurally stable ones. In this work the authors solve this problem completely modulo limit cycles and give all possible phase portraits for such structurally stable quadratic vector fields.