Long-read, single-molecule real-time (SMRT) sequencing is routinely used to finish microbial genomes, but available assembly methods have not scaled well to larger genomes. We introduce the MinHash Alignment Process (MHAP) for overlapping noisy, long reads using probabilistic, locality-sensitive hashing. Integrating MHAP with the Celera Assembler enabled reference-grade de novo assemblies of Saccharomyces cerevisiae, Arabidopsis thaliana, Drosophila melanogaster and a human hydatidiform mole cell line (CHM1) from SMRT sequencing. The resulting assemblies are highly continuous, include fully resolved chromosome arms and close persistent gaps in these reference genomes. Our assembly of D. melanogaster revealed previously unknown heterochromatic and telomeric transition sequences, and we assembled low-complexity sequences from CHM1 that fill gaps in the human GRCh38 reference. Using MHAP and the Celera Assembler, single-molecule sequencing can produce de novo near-complete eukaryotic assemblies that are 99.99% accurate when compared with available reference genomes.