In this Series paper, we review trends since the 2005 Lancet Series on Neonatal Survival to inform acceleration of progress for newborn health post-2015. On the basis of multicountry analyses and multi-stakeholder consultations, we propose national targets for 2035 of no more than 10 stillbirths per 1000 total births, and no more than 10 neonatal deaths per 1000 livebirths, compatible with the under-5 mortality targets of no more than 20 per 1000 livebirths.