The paper presents an ongoing research on the stress patterns of compounds and MWEs of the type ADJ+N and their corresponding free NPs in English and Bulgarian. The research focuses on the identification and the formal representation of the possible stress patterns of compounds and MWEs and free NPs. During our research so far, we have compiled a corpus of over 2000 compounds and MWEs, approx. 1000 for each language – English and Bulgarian. Our theoretical framework includes elements from different theories, i.e. the Generative Phonology Theory, the Metrical Theory, and the Theory of Primary accent first which all define the stress as a prosodic element. Our main goals are to specify the prosodic region where the stress is defined in English and Bulgarian MWEs and noun phrases and to define the main features of the stress in MWEs and free NPs in English and Bulgarian. The results of our research can serve for implementation into NLP modules for spoken language processing and generation.