This review evaluates instrumental variable (IV) estimators for Mendelian randomization (MR), a method using genetic variants as instruments to estimate causal effects of exposures on outcomes. IV analysis is used to infer causality from observational data, particularly in MR where genetic variants are used as instruments. The paper discusses various IV estimation methods, including the ratio method, two-stage methods, likelihood-based methods, and semi-parametric methods. It also addresses statistical inference, confidence intervals, and the use of multiple instruments, especially weak instruments. The paper emphasizes the importance of considering bias and coverage properties, particularly with weak instruments, and provides practical guidance on choosing appropriate methods. It also discusses the use of multiple instruments, including allele scores, and introduces a score-based approach for weak instruments. The paper concludes with a discussion of methodological issues related to the efficiency and validity of IV analyses, and the practical application of IV methods. The review highlights the importance of considering the assumptions underlying IV analysis, particularly in the context of MR, and provides a comprehensive comparison of different IV methods.This review evaluates instrumental variable (IV) estimators for Mendelian randomization (MR), a method using genetic variants as instruments to estimate causal effects of exposures on outcomes. IV analysis is used to infer causality from observational data, particularly in MR where genetic variants are used as instruments. The paper discusses various IV estimation methods, including the ratio method, two-stage methods, likelihood-based methods, and semi-parametric methods. It also addresses statistical inference, confidence intervals, and the use of multiple instruments, especially weak instruments. The paper emphasizes the importance of considering bias and coverage properties, particularly with weak instruments, and provides practical guidance on choosing appropriate methods. It also discusses the use of multiple instruments, including allele scores, and introduces a score-based approach for weak instruments. The paper concludes with a discussion of methodological issues related to the efficiency and validity of IV analyses, and the practical application of IV methods. The review highlights the importance of considering the assumptions underlying IV analysis, particularly in the context of MR, and provides a comprehensive comparison of different IV methods.