Explore the foundations of Intuitionistic Type Theory in this comprehensive lecture by Peter Dybjer, delivered as part of the Hausdorff Trimester Program: Types, Sets and Constructions. Delve into the historical context and key concepts of this mathematical framework, including types, applications, material systems, and type systems. Examine the role of recursion combinators, paper formats, and System E in the development of the theory. Investigate dependent types, existential quantification, and the concept of the universe within Intuitionistic Type Theory. Analyze free systems, the abortion of choice, and various models. Gain insights into the openness of this theoretical approach and its implications for mathematical reasoning and computation.