In the 1980s, following the classification of   finite simple groups, it was established that every finite simple group can be generated by two elements. A natural question arises: can we impose restrictions on these generators? Given a triple (a,b,c) of positive integers, we say that a finite group is an (a,b,c)-group if it can be generated by two elements of respective orders dividing a and b,  and having product of order dividing c. In other words, an (a,b,c)-group is a finite quotient of the triangle group 

T=T_{a,b,c}=\langle x,y,z: x^a=y^b=z^c=xyz=1 \rangle.

After a general introduction, we will present various results - some old, some  more recent - on finite simple quotients of triangle groups.