Given a surface, one can consider the set of free homotopy classes of oriented closed curves  (this is the set of equivalence classes ofmaps from the circle into 
 the surface, where two such maps are equivalent if the corresponding curves can be deformed one into the other.)Given a free homotopy class  one can ask what is 
 the minimum number of times (counted with multiplicity) in which a curve in that class intersects itself. This is the minimal self-intersection number of the free 
 homotopy class. Analogously, given two classes, one can ask what is the minimum number of times representatives of  these classes intersect. This is the minimal 
 intersection number of these two classes. In these talks, several problems (and some solutions) related to minimal intersection and self-intersection will be discussed. 
 Also, some  algebraic structures closely related to the intersection and and self-intersection will be introduced.