The construction of coverage intervals for a proportion is difficult, especially when the proportion is very small or very large. Most of the methods treated in the literature implicitly assume simple random sampling. These interval-construction methods are not immediately applicable to data derived from a complex sample design. Some recent papers have addressed this problem, proposing modifications for complex samples. Matters are further complicated when a one-sided coverage interval is desired. This article provides an extensive review of existing methods for constructing coverage intervals for a proportion under both simple random and complex sample designs. It also evaluates the empirical performances of different one-sided coverage intervals under both a simple random and a stratified random sample design.