The Grothendieck period conjecture (GPC) predicts that the transcendence degree of the field generated by the periods of a motive over a number field is equal to the dimension of the motivic Galois group of the motive.

Several important problems in transcendental number theory are special cases of the GPC. In particular, the GPC would imply that the set consisting of pi, the values of the Riemann zeta function at odd integers greater than 1 and the values of the logarithm function at prime numbers is algebraically independent. After going over a proof of this implication, we will try to make the argument more efficient. This leads us to consider motives whose motivic Galois groups have maximal unipotent radicals. The rest of the talk is on some recent results on the classification of such motives and the mysterious questions about periods that arise from these results. This talk is partially based on joint work with Kumar Murty.