In this paper, we study the received signal strength (RSS) based localization problem with correlated shadowing between pairs of RSS measurements. By linearizing the correlated RSS model, a weighted least squares (WLS) is formulated to obtain the target location. We also study the correlated shadowing when differential received signal strength (DRSS) is deployed as measurements. Numerical simulations show that the proposed algorithms outperform the algorithms that do not take the correlation between measurements into consideration.