Each human genome includes de novo mutations that arose during gametogenesis. While these germline mutations represent a fundamental source of new genetic diversity, they can also create deleterious alleles that impact fitness. The germline mutation rate for single nucleotide variants and factors that significantly influence this rate, such as parental age, are now well established. However, far less is known about the frequency, distribution, and features that impact de novo structural mutations. We report a large, family-based study of germline mutations, excluding aneuploidy, that affect genome structure among 572 genomes from 33 families in a multigenerational CEPH-Utah cohort and 2,363 cases of non-familial autism spectrum disorder (ASD), 1,938 unaffected siblings, and both parents (9,599 genomes in total). We find that de novo structural mutations detected by alignment-based, short-read WGS occurred at an overall rate of at least 0.160 events per genome in unaffected individuals and was significantly higher (0.206 per genome) in ASD cases. In both probands and unaffected samples, nearly 73% of de novo structural mutations arose in paternal gametes, and predict most de novo structural mutations to be caused by mutational mechanisms that do not require sequence homology. After multiple testing correction we did not observe a statistically significant correlation between parental age and the rate of de novo structural variation in offspring. These results highlight that a spectrum of mutational mechanisms contribute to germline structural mutations, and that these mechanisms likely have markedly different rates and selective pressures than those leading to point mutations.