We will study in this project exciting new interactions and applications between two fundamental modern research areas of mathematics, homotopy theory and higher category theory. These areas of mathematics are used in applied sciences. For instance, homotopy theory is used in robotics and in computer science. Higher category theory studies the way in which complex structures arising for instance in physics, computer science, biology, can be described by a common language, the one of ‘weak n-categories’. In this project, we apply ideas and techniques from homotopy theory to higher category theory. This will provide new and groundbreaking insights into the latter and will return homotopical applications. We will study certain structures which resembles simple algebraic ones but which are in fact much more complex because the defining data are specified ‘up to homotopy’. These structures are called homotopy algebras. We will then study ways in which a homotoy algebra can be made suitably equivalent to a simpler structure, a strict algebra. This process is called rigidification. We will then apply this theory to weak n-categories. We will view one of the models of weak n-categories, due to Tamsamani, as homotopy algebras, and study its rigidification. This will produce a new important type of higher categorical structure, called weakly globular n-fold categories. These will then be used in applications. We will obtain a new way to describe the building blocks of topological spaces, called n-types, and we will understand their connection with iterated loop spaces. We will also pursue other homotopical applications which will lead to the computation of important invariants used to describe topological spaces.