Abstract: Quantum computation and quantum cryptography are based on principles of quantum mechanics. In 1984, Bennett and Brassard proposed the first quantum key distribution protocol called BB84, which started the study of quantum cryptography. Since then, a great deal of work has been carried out in various fields such as quantum encryption and quantum signature. In 1994, Shor designed the first practical quantum algorithm which can factor large integers in polynomial time. Shor’s algorithm used Quantum Fourier Transform, which is the kernel of most quantum algorithms. In 1996, Grover designed a new algorithm which can search the unstructured data to get the required result in the time of approximately the square root of the total account of the data. Shor’s algorithm and Grover’s algorithm not only embody the advantages of quantum computing, but also pose a threat to the traditional cryptography based on mathematical difficulties such as RSA. After half a century’s development, quantum computing and quantum cryptography have achieved fruitful results in theory and practice. In this paper, we summarize the contents from the perspectives of the mathematical framework of quantum mechanics, basic concepts and principles, basic ideas of quantum computing, research progress and main ideas of quantum cryptography, etc.