The 20th century saw the development of many of the key concepts and theories in algebraic geometry. However, the evolution of style and approach over time has rendered the original texts challenging for modern readers to decipher. Bridging the gap between classical and modern algebraic geometry, this book explains classical results using modern tools and language. The second edition has undergone significant expansion. This first volume includes an extensive look at the enumerative geometry of quadrics and a more in-depth exploration of Cremona transformations, featuring more examples of different types. Furthermore, the expanded bibliography now encompasses over 800 references, including references to results obtained in the twelve years since the publication of the first edition. This carefully crafted reference will continue to keep classical algebraic geometry results alive and accessible to new generations of graduate students and researchers today.