Epistemic modality which deals with possibility (validity) and predictability is quite a complicated grammar topic for second language learners in comparison with deontic modality which indicates obligation, permission and desirability. While deontic modality refers to more evident attitudes, epistemic modality deals with subtle nuances of reality. The problem is aggravated by the fact that core English modal verbs (must, can, may, might, should, etc.) are duel and can express both types of modal meaning. Besides, a set of language means which express epistemic modality is rather extensive, and includes various adverbs and adjectives, e. g. possible, possibly, probably, perhaps, likely, unlikely, presumably, etc. The article aims to highlight difficulties in the acquisition of epistemic modality and suggest some teaching techniques to master it. The material for the linguistic analysis was obtained from British and American quality newspapers (The Guardian, Independent, The Times, The Economist, The New York Times, and The Washington Post) for the period 2018-2020. The paper argues that epistemic modality should be in the ESL teacher's focus not only at the grammar lessons but during the whole teaching process. Classes of translation might give a good opportunity to develop the relevant skills to discern epistemic modality and define its function in the text. In this paper I analyse newspaper texts and show that media writers often resort to epistemic modality as a means of creating ambiguity and uncertainty to articulate their ideas in an indirect and subtle way and thus to influence the public opinion. I argue that to understand the function of epistemic modals in the text and discover the author's message a thorough analysis of the text and context is needed. The combination of traditional teaching techniques with discourse-pragmatic analysis complemented by constructivist approach constitutes a relevant method of SL teaching and learning. It contributes to developing translation skills and educating independent thinkers.