Slice sampling is a Markov chain method for sampling from complex distributions by sampling uniformly from the region under the density function's plot. It involves alternating uniform sampling in the vertical and horizontal directions. This method can be applied to univariate and multivariate distributions, often being easier to implement than Gibbs sampling and more efficient than simple Metropolis updates due to its adaptive change magnitude selection. Slice sampling can adaptively choose changes based on local density properties and suppress random walks using 'overrelaxed' or 'effective' methods. It is particularly useful for hierarchical Bayesian models, where simple Metropolis methods may fail. Slice sampling methods can be implemented for single variables or all variables simultaneously, with the latter allowing for more efficient sampling by respecting variable dependencies. The paper discusses various slice sampling techniques, including single-variable and multivariate methods, and their advantages over traditional Markov chain methods like Gibbs sampling and Metropolis-Hastings. It also addresses the challenges of implementing these methods and provides proofs of correctness for single-variable slice sampling. The paper concludes with a discussion of the merits of slice sampling compared to other methods and its suitability for routine use.Slice sampling is a Markov chain method for sampling from complex distributions by sampling uniformly from the region under the density function's plot. It involves alternating uniform sampling in the vertical and horizontal directions. This method can be applied to univariate and multivariate distributions, often being easier to implement than Gibbs sampling and more efficient than simple Metropolis updates due to its adaptive change magnitude selection. Slice sampling can adaptively choose changes based on local density properties and suppress random walks using 'overrelaxed' or 'effective' methods. It is particularly useful for hierarchical Bayesian models, where simple Metropolis methods may fail. Slice sampling methods can be implemented for single variables or all variables simultaneously, with the latter allowing for more efficient sampling by respecting variable dependencies. The paper discusses various slice sampling techniques, including single-variable and multivariate methods, and their advantages over traditional Markov chain methods like Gibbs sampling and Metropolis-Hastings. It also addresses the challenges of implementing these methods and provides proofs of correctness for single-variable slice sampling. The paper concludes with a discussion of the merits of slice sampling compared to other methods and its suitability for routine use.