In this chapter we (really) briefly describe two important examples of “universal computers”: the Turing machine and its quantum counterpart, the quantum Turing machine. These “machines” are useful to check computability and efficiency of algorithms without specifying a particular hardware implementation, that is one of the main tasks of computer science. For the sake of completeness we also introduce the main complexity classes ( \(\mathsf {P}\) , \(\mathsf {NP}\) and their quantum analogue \(\mathsf {BQP}\) and \(\mathsf {QMA}\) ).