DescriptionWe study heat conduction and other nonequilibrium properties of one dimensional chain of particles, ordered and disordered, harmonic and anharmonic. Our results include derivation of oscillatory temperature profile for harmonic alternating mass chains, demonstration of finite heat conductivity for disordered harmonic chains with velocity-flipping noise, description of fluctuations for ordered harmonic systems with noise, and understanding the behavior of systems with noise and anharmonicity. We provide a comparative analysis of the effects of the various dynamics and system parameters on the characteristics of the nonequilibrium steady state of the chain.