Being able to extract from scientific papers their main points, key insights, and other important information, referred to here as aspects, might facilitate the process of conducting a scientific literature review. Therefore, the aim of our research is to create a tool for automatic aspect extraction from Russian-language scientific texts of any domain. In this paper, we present a cross-domain dataset of scientific texts in Russian, annotated with such aspects as Task, Contribution, Method, and Conclusion, as well as a baseline algorithm for aspect extraction, based on the multilingual BERT model fine-tuned on our data. We show that there are some differences in aspect representation in different domains, but even though our model was trained on a limited number of scientific domains, it is still able to generalize to new domains, as was proved by cross-domain experiments. The code and the dataset are available at \url{https://github.com/anna-marshalova/automatic-aspect-extraction-from-scientific-texts}.