Abstract: I will introduce the concept of dimension structures in abstract spaces. Given such a space endowed with a dimension structure one can define various characteristics of dimension type and study their basic properties. I will discuss three major examples of dimension structures:

1) The dimension structure associated with metric in a metric space; the corresponding dimension characteristics are the classical Hausdorff and box dimensions of sets and measures.

2) The dimension structure associated with a metric and a Borel measure in a metric space; the corresponding dimension characteristics are the dimension spectra including the well-known Hentschel-Procaccia dimension spectrum.

3) The dimension structure associated with a dynamical system and a continuous function in a compact metric space; the corresponding dimension characteristics are the topological pressure of the function; this reveals the dimension nature of the topological pressure as well as topological and metric entropies which are the classical invariants of dynamics.