Many exposures considered in Mendelian randomization (MR) studies are polygenic in that they are influenced by thousands of genetic variants. By using many single-nucleotide polymorphisms (SNPs) as instrumental variables, more variation in the exposure is explained, increasing the precision of MR. Furthermore, methods can be designed that relax the assumptions of MR, especially concerning direct pleiotropic effects on the outcome. This article reviews the concepts and assumptions underlying the commonly used polygenic MR methods. Using a polygenic score as an instrument is equivalent to a weighted mean of individual SNP results, and the other fundamental averages, median and mode, may also be used to estimate causal effects. Outlier detection is useful for identifying pleiotropic SNPs to be excluded from analysis. Bayesian approaches are available to incorporate prior beliefs about pleiotropy. These methods each entail different assumptions, and together provide a set of sensitivity analyses to help triangulate evidence about causality.