Structural variations are the greatest source of genetic variation, but they remain poorly understood because of technological limitations. Single-molecule long-read sequencing has the potential to dramatically advance the field, although high error rates are a challenge with existing methods. Addressing this need, we introduce open-source methods for long-read alignment (NGMLR; https://github.com/philres/ngmlr) and structural variant identification (Sniffles; https://github.com/fritzsedlazeck/Sniffles) that provide unprecedented sensitivity and precision for variant detection, even in repeat-rich regions and for complex nested events that can have substantial effects on human health. In several long-read datasets, including healthy and cancerous human genomes, we discovered thousands of novel variants and categorized systematic errors in short-read approaches. NGMLR and Sniffles can automatically filter false events and operate on low-coverage data, thereby reducing the high costs that have hindered the application of long reads in clinical and research settings.