Immersions of spheres into Euclidean spaces are classified up to regular homotopy by their Smale invariants. We shall discuss geometric formulas for Smale invariants primarily in dimensions right below double.
The set of self-transverse immersions is open and dense in the space of immersions. The regular homotopy classification of immersiosn is concerned with finding the path components of the space of immersions. We will consider the finer classification problem of finding the path components of the space of self-tansverse immersions. An important tool in this study are so called Vassiliev invariants. We show that in some dimensions Vassiliev invariants are too weak to distinguish the path components of the space of self-transverse immersions. In other cases the theory of Vassiliev invariants is much richer. We will describe what is known in these cases and state some open problems.