Young children aged birth to 5 years are known users of the internet, both unsupervised and in collaboration with adults. Adults also use the internet to share details of children's lives with others, via sharenting and educational apps. During COVID-19 internet use by children and families rose significantly during periods of enforced stay-home. Internet use by children, and by adults on behalf exposes children to conduct, contact and content risks online. These risks mean that cyber-safety in the early years is increasingly necessary, especially concerning increased internet usage during COVID-19. While cyber-safety is well developed for primary and secondary-school aged children this is not the case for young children, their families and educators. This paper proposes a research agenda for cyber-safety in the early years, using critical constructivism and internet studies to define the internet as a non-unitary technology. Three main objects of study concerning cyber-safety in the early years, including the reference to COVID-19 are identified for targeted research, including: technologies, context and policy.