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. We also give targets for 2030. Reduction of neonatal mortality has been slower than that for maternal and child (1-59 months) mortality, slowest in the highest burden countries, especially in Africa, and reduction is even slower for stillbirth rates. Birth is the time of highest risk, when more than 40% of maternal deaths (total about 290,000) and stillbirths or neonatal deaths (5·5 million) occur every year. These deaths happen rapidly, needing a rapid response by health-care workers. The 2·9 million annual neonatal deaths worldwide are attributable to three main causes: infections (0·6 million), intrapartum conditions (0·7 million), and preterm birth complications (1·0 million). Boys have a higher biological risk of neonatal death, but girls often have a higher social risk. Small size at birth--due to preterm birth or small-for-gestational-age (SGA), or both--is the biggest risk factor for more than 80% of neonatal deaths and increases risk of post-neonatal mortality, growth failure, and adult-onset non-communicable diseases. South Asia has the highest SGA rates and sub-Saharan Africa has the highest preterm birth rates. Babies who are term SGA low birthweight (10·4 million in these regions) are at risk of stunting and adult-onset metabolic conditions. 15 million preterm births, especially of those younger than 32 weeks' gestation, are at the highest risk of neonatal death, with ongoing post-neonatal mortality risk, and important risk of long-term neurodevelopmental impairment, stunting, and non-communicable conditions. 4 million neonates annually have other life-threatening or disabling conditions including intrapartum-related brain injury, severe bacterial infections, or pathological jaundice. Half of the world's newborn babies do not get a birth certificate, and most neonatal deaths and almost all stillbirths have no death certificate. To count deaths is crucial to change them. Failure to improve birth outcomes by 2035 will result in an estimated 116 million deaths, 99 million survivors with disability or lost development potential, and millions of adults at increased risk of non-communicable diseases after low birthweight. In the post-2015 era, improvements in child survival, development, and human capital depend on ensuring a healthy start for every newborn baby--the citizens and workforce of the future.