This volume is a collection of six lecture notes from the Abecedarian School of SIDE15 (ASIDE15-Milano-Italy), held before the 15th International Conference on Symmetries and Integrability of Difference Equations (SIDE15-Sirmione-Italy), Italy, in 2025. This book introduces contemporary research topics in discrete integrable systems, including applications of classical algebraic geometry, classical and modified Yang-Baxter equations, $r$-matrices, Fermi-Pasta-Ulam-Tsingou-type models, Hamiltonian approaches to differential-difference equations, and orthogonal polynomials with connections to discrete Painlevé equations. The focus lies on providing clear, self-contained expositions of both foundational and advanced ideas. The chapters explain key theoretical frameworks, give practical computational techniques, and emphasize important examples. Exercises are also included in some chapters. The book offers a wide range of perspectives for anyone seeking to learn or expand their research in discrete integrable systems and related fields. This book features both introductory and advanced topics, accessible exposition, and numerous insights into current research trends. It will be beneficial for graduate students and experienced mathematicians.