We propose sensitivity analyses for publication bias in meta-analyses. We consider a publication process such that “statistically significant” results are more likely to be published than negative or “nonsignificant” results by an unknown ratio, eta. Using inverse-probability weighting and robust estimation that accommodates non-normal true effects, small meta-analyses, and clustering, we develop sensitivity analyses that enable statements such as: “For publication bias to shift the observed point estimate to the null, ‘significant’ results would need to be at least 30-fold more likely to be published than negative or ‘nonsignificant’ results.” Comparable statements can be made regarding shifting to a chosen non-null value or shifting the confidence interval. To aid interpretation, we describe empirical benchmarks for plausible values of eta across disciplines. We show that a worst-case meta-analytic point estimate for maximal publication bias under the selection model can be obtained simply by conducting a standard meta-analysis of only the negative and “nonsignificant” studies; this method sometimes indicates that no amount of such publication bias could “explain away” the results. We illustrate the proposed methods using real-life meta-analyses and provide an R package, PublicationBias.