In this article, the Swedish findings from a European comparative study on 0-3-year-old children and their digital lives are presented and discussed in relation to domestication theory, including the concept of moral economy. More specifically, attention is paid to toddler's appropriation of digital technology and the parents' moral struggles: the negotiations between the parents concerning the introduction of digital media practices in early childhood, the selection of content, and the monitoring of children. Parents of very young children have ambivalent feelings towards digital media technologies and struggle to make the right decision for their children. The study demonstrates that the domestication of digital technology in early childhood is far more multifaceted and troublesome for parents to handle than previous research has found. © 2021 Helena Sandberg et al., published by Sciendo 2021.