Abstract: I will discuss about my recent joint work with Yu. Berest and F. Eshmatov on the structure of automorphism groups of class of algebras closely related to the first Weyl algebra $A_1(k)$. In particular, we give a geometric presentation for these groups using the Bass-Serre theory, thus answering question posed by T. Stafford in 80's. A key role in our approach is played by a transitive action of the automorphism group $Aut(A_1)$ on the Calogero-Moser varieties, which are certain variation of Hilbert schemes of points on the plane. In particular, using this geometric description of these groups we have been able to classify these groups up to isomorphism. The results obtained generalize some classic theorems of J. Dixmier and L. Makar-Limanov. In the end, we discuss a few open problems related to these groups such as the Dixmier Conjecture.