Transcriptome-wide association studies (TWAS) have proven to be a powerful tool to identify genes associated with human diseases by aggregating cis-regulatory effects on gene expression. However, TWAS relies on building predictive models of gene expression, which are sensitive to the sample size and tissue on which they are trained. The Gene Tissue Expression Project has produced reference transcriptomes across 53 human tissues and cell types; however, the data is highly sparse, making it difficult to build polygenic models in relevant tissues for TWAS. Here, we propose \textbackslashtextquoteleftfQTL\textbackslashtextquoteright, a multi-tissue, multivariate model for mapping expression quantitative trait loci and predicting gene expression. Our model decomposes eQTL effects into SNP-specific and tissue-specific components, pooling information across relevant tissues to effectively boost sample sizes. In simulation, we demonstrate that our multi-tissue approach outperforms single-tissue approaches in identifying causal eQTLs and tissues of action. Using our method, we fit polygenic models for 13,461 genes, characterized the tissue-specificity of the learned cis-eQTLs, and performed TWAS for Alzheimer's disease and schizophrenia, identifying 107 and 382 associated genes, respectively.