This book deals with the problem of bounding numerical invariants of nonsingular projective algebraic varieties from a unified point of view. Starting with a new insightful proof of the Castelnuovo theorem, the author proceeds with giving (sharp) bounds for basic numerical invariants (such as Betti, Hodge, and Chern numbers) of algebraic varieties of arbitrary dimension and classifying the varieties on the boundary. Many important results are proved in several different ways underlining different aspects of the topic. However, studying numerical invariants is just a pretext for the exploration of the rich and versatile interplay between geometry and topology of projective algebraic varieties which forms the core of the book. A special role in this study is played by the dual varieties and their degrees classically called classes. While making an extensive use of Lefschetz theory (which is now also available for varieties over an algebraically closed field of positive characteristic), the author also develops a new variant of Morse theory which yields statements (including bounds) on the level of CW-complexes and provides bounds for the homology of real algebraic varieties. In this way one also gets stronger forms of some classical results on the topology of projective varieties, such as weak and hard Lefschetz theorems, Hopf type formula etc.A detailed outline of the exciting centennial history of this topic and a list of intriguing open problems contribute to the value of the book.The exposition is accessible to students with only basic knowledge of algebraic geometry and topology.