An invitation to the positivity and geometry of algebraic cycles

The purpose of this work is an introduction and overview of geometric and numeric properties of algebraic cycles in smooth projective varieties. We recall or propose several problems, which we consider worth to study. We are mainly interested in, but do not restrict our story to, codimension 2 cycles in projective spaces. These are points in P^2, curves in P^3, surfaces in P^4 and so on