This paper develops asymptotic distribution theory for instrumental variable regression when the partial correlation between the instruments and a single included endogenous variable is weak, modeled as local to zero. Asymptotic representations are provided for various instrumental variable statistics, including the two-stage least squares (TSLS) and limited information maximum likelihood (LIML) estimators and their t-statistics. The asymptotic distributions are found to provide good approximations to sampling distributions with as few as 20 observations per instrument. Even in large samples, TSLS can be badly biased, but LIML is, in many cases, approximately median unbiased. The theory suggests concrete quantitative guidelines for applied work. These guidelines help to interpret Angrist and Krueger's (1991) estimates of the returns to education: whereas TSLS estimates with many instruments approach the OLS estimate of 6%, the more reliable LIML and TSLS estimates with fewer instruments fall between 8% and 10%, with a typical confidence interval of (6%, 14%). The paper also discusses the asymptotic behavior of TSLS and LIML estimators under weak instruments, the distribution of overidentifying restrictions tests, and the asymptotic distribution of LIML. The results suggest that LIML point estimates are approximately median-unbiased and LIML confidence intervals have approximately their nominal coverage rates, even for weak instruments. The paper concludes that key features of the distributions, such as bias and coverage rates, can be summarized in a series of simple plots which are applicable to a wide range of models and number of instruments. These plots can be used to provide concrete quantitative guidelines for empirical applications of instrumental variables regressions.This paper develops asymptotic distribution theory for instrumental variable regression when the partial correlation between the instruments and a single included endogenous variable is weak, modeled as local to zero. Asymptotic representations are provided for various instrumental variable statistics, including the two-stage least squares (TSLS) and limited information maximum likelihood (LIML) estimators and their t-statistics. The asymptotic distributions are found to provide good approximations to sampling distributions with as few as 20 observations per instrument. Even in large samples, TSLS can be badly biased, but LIML is, in many cases, approximately median unbiased. The theory suggests concrete quantitative guidelines for applied work. These guidelines help to interpret Angrist and Krueger's (1991) estimates of the returns to education: whereas TSLS estimates with many instruments approach the OLS estimate of 6%, the more reliable LIML and TSLS estimates with fewer instruments fall between 8% and 10%, with a typical confidence interval of (6%, 14%). The paper also discusses the asymptotic behavior of TSLS and LIML estimators under weak instruments, the distribution of overidentifying restrictions tests, and the asymptotic distribution of LIML. The results suggest that LIML point estimates are approximately median-unbiased and LIML confidence intervals have approximately their nominal coverage rates, even for weak instruments. The paper concludes that key features of the distributions, such as bias and coverage rates, can be summarized in a series of simple plots which are applicable to a wide range of models and number of instruments. These plots can be used to provide concrete quantitative guidelines for empirical applications of instrumental variables regressions.