High level of interpersonal and institutional trust facilitates interactions between people, reduces transaction costs, and stimulates economic activity. Trust becomes especially important during crises when people are required to act together and to follow recommendations. The coronavirus pandemic has once again made this clear. Using data from Eurofund, we expose the relationship between various components of trust (public authorities, healthcare systems, science, etc.) and behavior of the Europeans during the COVID-19 pandemic (willingness to comply with quarantine requirements and get vaccinated). Intention to get vaccinated is most closely related to trust in science and in the healthcare system, while the willingness to comply with sanitary regime is most closely related to trust in government. Readiness to comply with epidemiological rules and willingness to get vaccinated vary greatly between countries. In the countries with higher institutional and interpersonal trust people express greater willingness to follow epidemiological recommendations announced by authorities. We selected two clusters from our sample (with lower and higher levels of trust) and chose Italy and Sweden as representatives of these clusters to illustrate differential responses to sanitary restrictions imposed by government agencies. In Italy’s case, ambiguous attitude, and in Sweden’s case, almost complete acceptance resulted in different levels of COVID-19 mortality. At the same time, both countries have achieved relatively high vaccination rates, which reflects reliance on science and reliability of national health systems.