Length-biased distributions arise when the probability of including an observational unit in a sample is proportional to its length. They form an important special case of weighted distributions and have substantial applications in survival analysis, renewal processes, reliability studies, biomedical investigations, ecology, actuarial science and related areas. The present review provides a concise but systematic account of the theory and development of length-biased distributions, with emphasis on their mathematical formulation, historical development, statistical properties, inferential methods and applications. Special attention is given to models used in lifetime data analysis, including the Weibull length-biased distribution, Weibull length-biased exponential distribution, length-biased beta-Pareto distribution, length-biased weighted Lomax distribution, length-biased Sushila distribution, length-biased Aradhana distribution and length-biased transmuted models. The review also discusses nonparametric estimation, goodness-of-fit procedures, maximum likelihood and Bayesian estimation, and the comparative performance of length-biased models in lifetime data analysis. The study also classifies the existing literature and identifies potential areas for future research.