In 1982 Richard Feynman (Int J Theor Phys 21:467, 1982, [1]) suggested there were certain problems that would be difficult to preform on a computer running according to classical mechanics but which would be easy to do on a computer running according to quantum principles. The reason why this is so is easy to see. A quantum system consisting of N interacting spins requires a simulation using vectors of \(2^N\) dimensions in general. This exponential growth of the basis size is what makes classical simulations of complex quantum problems so difficult. On the other hand if we built a system with N interacting spins and allowed it to evolve unitarily, no such difficulty would be encountered. It would appear that a computer executing unitary evolution on a system of two level systems could significantly outperform a classical computer set to solve the equivalent problem. In this chapter we introduce the idea of universal quantum computation and discuss various quantum optical schemes for implementing it.