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Amino acid insertions and deletions (indels) are an abundant class of genetic variants. However, compared to substitutions, the effects of indels on protein stability are not well understood and are poorly predicted. To better understand indels here we analyze new and existing large-scale deep indel mutagenesis (DIM) of structurally diverse proteins. The effects of indels on protein stability vary extensively among and within proteins and are not well predicted by existing computational methods. To address this shortcoming we present INDELi, a series of models that combine experimental or predicted substitution effects and secondary structure information to provide good prediction of the effects of indels on both protein stability and pathogenicity. Moreover, quantifying the effects of indels on protein-protein interactions suggests that insertions can be an important class of gain-of-function variants. Our results provide an overview of the impact of indels on proteins and a method to predict their effects genome-wide.