In today's digital society, personalization has become a crucial aspect of software applications, significantly impacting user experience and engagement. A new wave of intelligent user interfaces, such as AI-based conversational agents, has the potential to enable such personalization beyond what other types of interfaces could offer in the past. Personalization requires the ability to specify a complete user profile, covering as many dimensions as possible, such as potential accessibility constraints, interaction preferences, and even hobbies. In this sense, this paper presents the concepts of a unified user modeling language, aimed to combine previous approaches in a single proposal. Additionally, a proof of concept has been developed that leverages user profiles modeled using our language to automatically adapt a conversational agent.