AI for software engineering has made remarkable progress, becoming a notable success within generative AI. Despite this, achieving fully automated software engineering is still a significant challenge, requiring research efforts across both academia and industry. In this position paper, our goal is threefold. First, we provide a taxonomy of measures and tasks to categorize work towards AI software engineering. Second, we outline the key bottlenecks permeating today's approaches. Finally, we highlight promising paths towards making progress on these bottlenecks to guide future research in this rapidly maturing field.