This paper investigates two classes of relay channels, the Gaussian relay channel and the Gaussian two-way relay channel, when additive noises at the relay and destination(s) are correlated. Lattice codes are used to achieve the rate region for Compress-and-Forward (relay channel) and Compress/Decode-and-Forward (two-way relay channel). Numerical calculations show that there exist particular values of the correlation coefficient such that the gap between the Cut-Set Bound (CSB) and the proposed schemes is minimal.