Zika virus (ZIKV) infection was historically considered a disease with mild symptoms and no major consequences to human health. However, several long-term, late onset, and chronic neurological complications, both in congenitally-exposed babies and in adult patients, have been reported after ZIKV infection, especially after the 2015 epidemics in the American continent. The development or severity of these conditions cannot be fully predicted, but it is possible that genetic, epigenetic, and environmental factors may contribute to determine ZIKV infection outcomes. This reinforces the importance that individuals exposed to ZIKV are submitted to long-term clinical surveillance and highlights the urgent need for the development of therapeutic approaches to reduce or eliminate the neurological burden of infection. Here, we review the epidemiology of ZIKV-associated neurological complications and the role of factors that may influence disease outcome. Moreover, we discuss experimental and clinical evidence of drugs that have shown promising results in vitro or in vitro against viral replication and and/or ZIKV-induced neurotoxicity.