Viscosity solutions to degenerate Complex Monge-Amp\`ere equations

We develop an alternative approach to Degenerate complex Monge-Amp\`ere equations on compact K\"ahler manifolds based on the concept of viscosity solutions and compare systematically viscosity concepts with pluripotential theoretic ones. We generalize to the K\"ahler case a theorem due to Dinew and Zhang in the projective case to the effect that their pluripotential solutions constructed previously by the authors are continuous.