I will discuss the notion of weakly commensurable Zariski-dense
subgroups of semi-simple groups. This notion was introduced in my
joint work with Andrei Rapinchuk (Publ. Math. IHES 109(2009), 113-184),
where we determined when two Zariski-dense S-arithmetic subgroups of
absolutely almost simple algebraic groups over a field of characteristic
zero are weakly commensurable. These results enabled us to prove that in
many situations isospectral locally symmetric spaces of simple real
algebraic groups are necessarily commensurable. This settled the famous
question "Can one hear the shape of a drum?" of Mark Kac for these spaces.