Studying the functional consequences of structural variants (SVs) in mammalian genomes is challenging because (i) SVs arise much less commonly than single-nucleotide variants or small indels and (ii) methods to generate, map, and characterize SVs in model systems are underdeveloped. To address these challenges, we developed Genome-Shuffle-seq, a method that enables the multiplex generation and mapping of thousands of SVs (deletions, inversions, translocations, and extrachromosomal circles) throughout mammalian genomes. We also demonstrate the co-capture of SV identity with single-cell transcriptomes, facilitating the measurement of SV impact on gene expression. We anticipate that Genome-Shuffle-seq will be broadly useful for the systematic exploration of the functional consequences of SVs on gene expression, the chromatin landscape, and three-dimensional nuclear architecture, while also initiating a path toward a minimal mammalian genome.