Homotopy theory is a branch of mathematics that studies the properties of topological spaces that are preserved under continuous transformations. It focuses on the concept of homotopy, which is a way to deform one continuous function into another without breaking or tearing. This allows mathematicians to classify spaces based on their shape and connectivity. In homotopy theory, two spaces are considered equivalent if they can be transformed into each other through a series of continuous deformations. This leads to the development of important concepts such as homotopy groups, homotopy equivalence, and simplicial complexes, which help in understanding the underlying structure of spaces in a more abstract way.