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This paper reviews two main types of prediction interval methods under a parametric framework. First, we describe methods based on an (approximate) pivotal quantity. Examples include the plug-in, pivotal, and calibration methods. Then we describe methods based on a predictive distribution (sometimes derived based on the likelihood). Examples include Bayesian, fiducial, and direct-bootstrap methods. Several examples involving continuous distributions along with simulation studies to evaluate coverage probability properties are provided. We provide specific connections among different prediction interval methods for the (log-)location-scale family of distributions. This paper also discusses general prediction interval methods for discrete data, using the binomial and Poisson distributions as examples. We also overview methods for dependent data, with application to time series, spatial data, and Markov random fields, for example.