This book provides a highly accessible approach to discrete surface theory, within the unifying frameworks of Moebius and Lie sphere geometries, from the perspective of transformation theory of surfaces rooted in integrable systems.  It elucidates how the transformation theory for smooth surfaces can be used as a springboard for understanding the discretization process of certain types of surfaces, and it is aimed at high-level undergraduate students, graduate students and professional mathematicians alike.  The reader will benefit from the detailed exploration of the transformation theory of surfaces, including Christoffel, Calapso and Darboux transformations of particular classes of surfaces, as well as becoming more familiar with integrable systems via zero curvature representation, including flat connections and conserved quantities, in both smooth and discrete settings.