This talk describes a systematic approach to generating discrete time stepping meth ods from energy functions (Hamiltonians) which exactly conserve the energy, and often other conserved quantities of the corresponding continuous evolution equations, through respecting continuous and discrete symmetries of the Hamiltonian.These serve as nu merical methods for Hamiltonian DEs but are also of intrinsic interest as totally discrete conservative dynamical systems with symmetries such as shifts around a spatial period.In particular, this is applied to the case of the completely integrable focusing cubic NLS with periodic boundary conditions, via spatial semi-discretizations including the AblowitzLadik system.