Availability of powerful computation and communication technology as well as advances in artificial intelligence enable a new generation of complex, AI-intense systems and applications. Such systems and applications promise exciting improvements on a societal level, yet they also bring with them new challenges for their development. In this paper we argue that significant challenges relate to defining and ensuring behaviour and quality attributes of such systems and applications. We specifically derive four challenge areas from relevant use cases of complex, AI-intense systems and applications related to industry, transportation, and home automation: understanding, determining, and specifying (i) contextual definitions and requirements, (ii) data attributes and requirements, (iii) performance definition and monitoring, and (iv) the impact of human factors on system acceptance and success. Solving these challenges will imply process support that integrates new requirements engineering methods into development approaches for complex, AI-intense systems and applications. We present these challenges in detail and propose a research roadmap.