Quantum computing is a rapidly advancing field with the potential to revolutionize computation. This paper provides an overview of quantum computing, emphasizing its mathematical foundations and practical implications. We discuss key concepts from quantum mechanics that form the basis of quantum computing, such as superposition and entanglement, and explore quantum algorithms like Shor's algorithm and Grover's algorithm. The paper also examines the practical implications of quantum computing in cryptography, optimization, and machine learning, highlighting quantum key distribution, quantum annealing, and quantum neural networks. Furthermore, we discuss the challenges and future directions of quantum computing, including error correction, scalability, and achieving quantum supremacy. Addressing these challenges will pave the way for realizing the full potential of quantum computing and unlocking new possibilities in computation and simulation.