Gaussian isokinetic and isoenergetic deterministic thermostats are reviewed in the correct historical context with their later justification using Gauss' principle of least constraint. The Nose-Hoover thermostat for simulating the canonical ensemble is also developed. For some model systems the Lyapunov exponents satisfy the conjugate pairing rule and a Hamiltonian formulation is obtained. We prove the conjugate pairing rule for nonequilibrium systems where the force is derivable from a potential. The generalized symplectic structure and Hamiltonian formulation is discussed. The application of such thermostats to the Lorentz gas is considered in some detail. The periodic orbit expansion methods are used to calculate averages and to categorize the generic transitions in the structure of the attractor. We prove that the conductivity in the nonequilibrium Lorentz gas is non-negative. (c) 1998 American Institute of Physics.