In this contributed talk we introduce, in a theoretical representation setting, a necessary and sufficient condition, namely the Rockland condition, for a left-invariant differential operator on a compact Lie group G to be globally hypoelliptic. In particular, we focus on the case of a product of two compact Lie groups G=G1×G2 and we show some examples on T^2 and on T^1×SU(2). It is possible to prove the existence of globally hypoelliptic smooth-coefficient operators that are not locally hypoelliptic. In the end, we present a class of pseudodifferential operators on the product G=G1×G2 and the so called bisingular pseudodifferential calculus, as introduced by L. Rodino in 1975.