The development of new computational techniques and better computing power has made it possible to attack some classical problems of algebraic geometry. The main goal of this book is to highlight such computational techniques related to algebraic curves. The area of research in algebraic curves is receiving more interest not only from the mathematics community, but also from engineers and computer scientists, because of the importance of algebraic curves in applications including cryptography, coding theory, error-correcting codes, digital imaging, computer vision, and many more. This book covers a wide variety of topics in the area, including elliptic curve cryptography, hyper elliptic curves, representations on some Riemann-Roch spaces of modular curves, computation of Hurwitz spectra, generating systems of finite groups, and Galois groups of polynomials, among other topics.