Since the explosive outbreak of Zika virus in Brazil and South/Central America in 2015–2016, the frequency of infections has subsided, but Zika virus remains present in this region as well as other tropical and sub-tropical areas of the globe. The most alarming aspect of Zika virus infection is its association with severe birth defects when infection occurs in pregnant women. Understanding the mechanism of Zika virus pathogenesis, which comprises features unique to Zika virus as well as shared with other teratogenic pathogens, is key to future prophylactic or therapeutic interventions. Nonhuman primate-based research has played a significant role in advancing our knowledge of Zika virus pathogenesis, especially with regard to fetal infection. This review summarizes what we have learned from these models and potential future research directions.