The information landscape has undergone dramatic changes with the expansion of the internet and digital social media platforms. Information can be spread more rapidly and can reach more people than ever before. While this offers excellent possibilities to teach and educate and to disseminate information about research results and scientific evidence, it also comes with a downside: False information can be propagated with equal ease and speed. This discussion paper describes and discusses the problems and the consequences of science disinformation in three areas of concern, namely climate change, vaccines and pandemics, and what we can do to increase awareness and minimize harm caused by the spread of disinformation. It does so by highlighting the societal value of the scientific method, research integrity, open science communication and the resulting trust in science. The underlying question is how to protect the pillars of science from the severe consequences of disinformation while maintaining openness and democratic principles. This paper presents the central characteristics of science disinformation, its roots, its spread, and potential solutions. The mere existence of disinformation is hard to prevent in open societies with strong protection of individual rights and freedom of expression. The paper identifies underlying cognitive, social and economic mechanisms that amplify the spread of disinformation and evaluates potential solutions. Extensive research over the past several years has identified cognitive features of the human mind, as well as fast and efficient transmission channels, that contribute to the prevalence of science disinformation in our societies. Potential solutions cover a range of psychological, technical and political measures including inoculation, debunking, recommender systems, fact-checking, raising awareness, media literacy, and innovations in science communication and public engagement. Together, they contribute to tackling problems such as knowledge resistance, pseudoscience, undermining of trust, confirmation bias, filter bubbles, echo chambers, and other problems related to science disinformation.