  We define integral geometric analogues of the Chern classes for real vector
bundle on a smooth real variety. More precisely, we define the Chern densities
of a real bundle. These densities are analogues of the Chern forms of a complex
vector bundle and inherit some of their properties.
  \noindent (The text is a summary of a report on the conference PCA'2024 in
Euler International Mathematical Institute, St. Petersburg)
