Kolmogorov-Arnold Networks (KANs), whose design is inspired-rather than dictated-by the Kolmogorov superposition theorem, have emerged as a structured alternative to MLPs. This review provides a systematic and comprehensive overview of the rapidly expanding KAN literature. The review is organized around three core themes: (i) clarifying the relationships between KANs and Kolmogorov superposition theory (KST), MLPs, and classical kernel methods; (ii) analyzing basis functions as a central design axis; and (iii) summarizing recent advances in accuracy, efficiency, regularization, and convergence. Finally, we provide a practical "Choose-Your-KAN" guide and outline open research challenges and future directions. The accompanying GitHub repository serves as a structured reference for ongoing KAN research.