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Much like our everyday moments of novelty, innovations often catalyse subsequent breakthroughs, creating a chain of progress that can be traced across history. The Science of the New explores the dynamics of novelty and innovation. It originates from the observation that these two interrelated phenomena that shape human experience, from the ordinary encounters we have in daily life to the groundbreaking advancements in science, technology, and society, are two faces of the same medal. In other words, novelties and innovation can be described by the same underlying processes and their emergence obeys the same general statistical laws. The construction of a solid mathematical framework to treat the phenomenologies related to the experience of the new represents a formidable challenge and a necessary one if we want to understand the world around us and be able to make predictions in an ever-changing environment and society. This book provides a minimal toolkit to face this problem. Organised into three distinct parts, it begins by outlining the fundamental theoretical tools necessary for analysing how novelties and innovations emerge. It then delves into classical and contemporary models that explain the processes behind innovation, illustrating the deep relationship between several ideas that took birth in different fields and influenced one another. The final section provides empirical case studies, applying the discussed frameworks to real-world systems and showing how mathematical and computational methods can help us understand innovation in various contexts.