To achieve sustainable development worldwide, the United Nations set 17 Sustainable Development Goals (SDGs) for humanity to reach by 2030. Society is involved in the challenge, with firms playing a crucial role. Thus, a key question is to what extent firms engage with the SDGs. Efforts to map firms' contributions have mainly focused on analysing companies' reports based on limited samples and non-real-time data. We present a novel interdisciplinary approach based on analysing big data from an online social network (Twitter) with complex network methods from statistical physics. By doing so, we provide a comprehensive and nearly real-time picture of firms' engagement with SDGs. Results show that: 1) SDGs themes tie conversations among major UK firms together; 2) the social dimension is predominant; 3) the attention to different SDGs themes varies depending on the community and sector firms belong to; 4) stakeholder engagement is higher on posts related to global challenges compared to general ones; 5) large UK companies and stakeholders generally behave differently from Italian ones. This paper provides theoretical contributions and practical implications relevant to firms, policymakers and management education. Most importantly, it provides a novel tool and a set of keywords to monitor the influence of the private sector on the implementation of the 2030 Agenda.