Quantum computing, a paradigm-shifting technology, leverages the principles of quantum mechanics to perform computations far beyond the capabilities of classical systems. This chapter provides a comprehensive overview of quantum computing algorithms, emphasizing their theoretical foundations, design, and practical applications. It begins by exploring fundamental quantum principles, such as superposition, entanglement, and quantum interference, which serve as the basis for quantum computation. Key algorithms are analysed, including Shor’s algorithm for integer factorization, Grover’s algorithm for database search, and the Quantum Approximate Optimization Algorithm (QAOA) for solving combinatorial problems. The chapter also introduces the growing field of hybrid quantum–classical algorithms, highlighting their potential for addressing real-world optimization, machine learning, and cryptography challenges. Implementation challenges, such as error correction and the scalability of quantum hardware, are discussed to provide a balanced perspective on the current state of the field. Finally, the chapter will discuss future trends in quantum algorithm research, including advancements in quantum error mitigation and the potential integration of quantum algorithms into existing computational frameworks. This chapter will serve as a resource for researchers, practitioners, and students aiming to understand and contribute to the evolving landscape of quantum computing algorithms.