The notion 'discourse competence' is widely used in linguistics, however, it has not received sufficient attention within the discipline of 'Russian as a Foreign Language', although it is obvious that direct usage of the techniques developed for the foreign languages, e.g., English, in particular, is impossible while teaching and/or learning Russian. The aim of this work is to describe the main features of discourse competence as a linguistic category in the context of the discipline 'Russian as a Foreign Language', to identify the main factors influencing its development and to compare the process of constructing a text/discourse by a foreigner and a native speaker. Ontologically discourse competence is a part of verbal-logical thinking, while from the methodological point of view it is a collection of certain language skills which makes it possible to construct a text. We believe that discourse competence requires competence in the following areas: (1) formal logic, (2) stylistics, and (3) textual genre, each of which plays an important role in constructing coherent and cohesive text. The way how discourse is being constructed depends on several factors such as (1) level of proficiency in the (Russian) language, (2) goals and objectives of the specific language course attended by a student, (3) nationality and cultural background of a student, and (4) cognitive style and individual (personality) traits of a student. We suggest using authentic sources starting from the elementary level while developing the discourse competence using the following model: reading an authentic text, copying and reproducing it, and, finally, constructing a text of one's own. The study materials should be prepared taking into consideration cultural background as well as cognitive style of the students.