Data Science and Machine Learning (ML) have become indispensable tools for extracting knowledge and insights from the ever-growing volume of digital data. The success of ML algorithms heavily depends on mathematical principles such as linear algebra, probability theory, statistics, and optimization. These mathematical foundations enable models to identify patterns, make predictions, and facilitate decision-making across various domains. This paper explores the key mathematical concepts that underpin Data Science and Machine Learning, the major categories of ML algorithms, and their real-world applications in fields like healthcare, finance, and climate science. Furthermore, it discusses emerging challenges such as interpretability, data bias, ethical considerations, and computational complexity, and outlines promising directions for future research.