Modeling certain phenomena using geometry sometimes requires the introduction of geometric structures on the spaces under consideration. We are interested here in contact structures. These naturally appear in control theory, in geometric optics, as well as in thermodynamics. The motion of a skater on the ice rink can be described using an abstract, three-dimensional space. The constraints in the motion of the skater are described by a contact structure on this abstract space. Contact geometry then provides a better understanding of the possible displacements of the skater. This approach can also be applied to other practical situations, such as parallel parking. Modeling geometric optics using contact geometry leads to the unification between two different descriptions of light, as a collection of light rays (Fermat’s principle) and as a wave (Huygens’ principle). The propagation of light in media with varying optical properties illustrates the link between contact geometry and dynamical systems. Contact geometry is a relatively recent field, but it is the object of very intensive research. The corresponding terminology started to appear near the end of the 19th century, and then important developments started and intensified during the second half of the 20th century. Despite important research results, many fundamental questions remain unanswered.