In the previous chapters we addressed quantum computation considering the quantum circuit model, where the information, encoded into qubits, is processed by means of quantum gates implementing a well defined algorithm. In this chapter we introduce a different approach to quantum computation, based on the adiabatic evolution. Here the problem is encoded into a given problem Hamiltonian and the solution is given by the ground state of the Hamiltonian itself. In order to find the solution, one starts form the ground state of an initial Hamiltonian which is then adiabatically transformed into the problem Hamiltonian. If the requirement of the so-called adiabatic theorem are satisfied, during the dynamics the system remains into the ground state of the instantaneous Hamiltonian and, thus, we finally end up in the ground state of the problem Hamiltonian.