The study of the geometry of dynamically defined fractals has advanced significantly over the past 60 years. A central element of this theory is the concept of horseshoes. These hyperbolic sets arise naturally in chaotic systems and exhibit remarkable fractal structures. Most studies on the geometric aspects of dynamically defined sets focus primarily on the cases of surfaces dynamics. As expected, the situation becomes more complex in higher dimensions. In this talk, we will discuss some basic questions that remain unanswered in the theory of higher dimensional horseshoes. Additionally, we will explore how combinatorial tools can be utilised to extract properties of typical horseshoes without significantly losing their complexity. This is a joint work with Carlos G. Moreira.