This book highlights and addresses a crucial need in the emerging field of Scientific Machine Learning (SciML) by offering a comprehensive and accessible guide that blends theory, algorithms, and applications. It explores how the synergy between machine learning and scientific computing can lead to more accurate, interpretable, and efficient models for scientific discovery. The book covers foundational mathematical principles, physics-informed neural networks, optimization techniques, uncertainty quantification, deep learning for scientific data, transformer-based foundational models, and neuro-symbolic reasoning. By combining domain knowledge with modern AI, SciML opens new frontiers in disciplines such as physics, biology, and engineering. This book is an essential resource for students, researchers, and professionals aiming to apply AI in scientific domains.