Climate changepoint (homogenization) methods abound today, with a myriad of techniques existing in both the climate and statistics literature. Unfortunately, the appropriate changepoint technique to use remains unclear to many. Further complicating issues, changepoint conclusions are not robust to small perturbations in assumptions; for example, allowing for a trend or correlation in the series can drastically change conclusions. This paper is a review of the changepoint topic, with an emphasis on illuminating the models and techniques that allow the scientist to make reliable conclusions. Pitfalls to avoid are demonstrated via actual applications. The discourse begins by narrating the salient statistical features of most climate time series. Thereafter, single and multiple changepoint problems are considered. Several pitfalls are discussed en route and good practices are recommended. While the majority of our applications involve temperature series, other settings are mentioned.