This paper develops asymptotic distribution theory for single-equation instrumental variables regression when the partial correlations between the instruments and the endogenous variables are weak, here modeled as local to zero. Asymptotic representations are provided for various statistics, including two-stage least squares (TSLS) and limited information maximum likelihood (LIML) estimators, Wald statistics, and statistics testing overidentification and endogeneity. The asymptotic distributions are found to provide good approximations to sampling distributions with 10-20 observations per instrument. The theory suggests concrete guidelines for applied work, including using nonstandard methods for construction of confidence regions. These results are used 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 estimates with fewer instruments fall between 8% and 10%, with a typical 95% confidence interval of (5%, 15%).