A remarkable application of quantum mechanical concepts of coherent superposition and quantum entanglement is a quantum computer which can solve certain problems at speeds unbelievably faster than the conventional computer. In this chapter, the basic principles and the conditions for the implementation of the quantum computer are introduced and the limitations imposed by the probabilistic nature of quantum mechanics and the inevitable decoherence phenomenon are discussed. Next the basic building blocks, the quantum logic gates, are introduced. These include the Hadamard, the CNOT, and the quantum phase gates. After these preliminaries, the implementation of the Deutsch algorithm, quantum teleportation, and quantum dense coding in terms of the quantum logic gates are discussed. It is also shown how the Bell states can be produced and measured using a sequence of quantum logic gates.