Human-AI co-creativity involves a human and an AI collaborating as partners on creative tasks such as generating music or art. This research domain is particularly timely as AI becomes increasingly prevalent in collaborative spaces. With the availability of ChatGPT, DALL.E 2 and other generative AI tools, co-creative AI is gaining popularity. Unlike general human-computer interaction, human-AI co-creation establishes a complex relationship where AI actively contributes, assumes human-like roles, and generates novel content blended with the user's contribution. Therefore, designing engaging and ethical co-creative systems poses challenges due to the open-ended nature of human-AI interaction. This dissertation contributes empirically and theoretically to the design of engaging and ethical human-centered co-creative AI. It focuses on four main areas: designing interaction, the impact of AI-to-human communication, ethical guidelines and understanding users' mental models of co-creative AI in human-AI co-creation. Firstly, this dissertation introduces the Co-Creative Framework for Interaction Design (COFI), which describes the broad range of possibilities for designing interactions in co-creative AI. Additionally, an analysis of 92 existing co-creative AI systems identifies common interaction design trends and research gaps. The analysis reveals a notable gap in commonly employed interaction designs: the absence of two-way communication between humans and AI, where AI cannot communicate with humans, limiting their potential as partners. Inspired by the research gap identified, this dissertation delves into examining the impact of AI-to-human communication on user experience and perception of co-creative AI. Two prototypes of a co-creative system, with and without AI-to-human communication, were developed to facilitate a comparative study. The results show an improved collaborative experience and user engagement with the AI that can communicate. Moreover, the results shed light on emerging ethical concerns alongside increased user engagement. This dissertation further explores the ethical challenges in human-AI co-creation by taking a human-centered approach. A design fiction study is presented to explore several ethical dilemmas and challenges in human-AI co-creation from the perspective of potential users. Findings provide potential users' perspectives, stances, and expectations, serving as a foundation for designing human-centered ethical AI partners in human-AI co-creation. Finally, this dissertation investigates users' mental models of co-creative AI, which is essential to understand designing human-centered co-creative AI. Through a survey study, we delve into users' mental models of co-creative AI and their association with user demographics to identify ways to design value-sensitive co-creative AI. The results obtained also lay the groundwork for future research on personalization in the realm of human-AI co-creation. The findings and frameworks presented lay the groundwork for future advancements in the field of human-AI co-creativity.