The digitalization of societies, accelerated by the COVID-19 pandemic, is an unstoppable process. This paper seeks to answer the question: what post-COVID-19 digital competences are needed for a sustainable society? It also aims to analyze the digitalization processes in education for shaping a sustainable digital society. A bibliographic search was performed on some of the most relevant international databases of scientific literature and the selected documents were analyzed through a content analysis. It is concluded that digital education has experienced a strong increase, reinforced by COVID-19, shaping the digital presence in all dimensions of life. However, it is not sufficient to assume that the new generations are naturally engaged in and can master digital social sustainability. The results demonstrate the importance of literacy and the unavoidable promotion of sustainability in a digital society. However, this digitalization of the educational process poses several challenges: it requires both software and hardware conditions, as well as digital literacy as a result of a complex of literacies. It also implies that teachers and students change their standpoints and practices with the attainment of new teaching and learning competences in order to fight the digital divide and to foster the widest possible social inclusion for the promotion of sustainable society—digitainability.