Numerous records from the North Atlantic and the surrounding continents have shown rapid and large amplitude climate variability during the last glacial period. This variability has often been associated to changes in the Atlantic Meridional Overturning Circulation (AMOC). Rapid climate change on the same time scales has also been reconstructed for sites far away from the North Atlantic, such as the tropical Atlantic, the East Pacific and Asia. The mechanisms explaining these climatic responses to the state of the AMOC are far from being completely understood, especially in a glacial context. Here we study three glacial simulations characterised by different AMOC strengths: 18, 15 and 2 Sv. With these simulations, we analyse the global climate sensitivity to a weak (18 to 15 Sv) and a strong (15 to 2 Sv) decrease in the AMOC strength. A weak decrease in the AMOC is associated, in our model simulations, to the classical North Atlantic and European cooling, but this cooling is not homogeneous over this region. We investigate the reasons for a lesser cooling (or even slight warming in some cases) over the Norwegian Sea and Northwestern Europe. It appears that the convection site in this area is active in both simulations, but that convection is unexpectedly stronger in the 15 Sv simulation. Due to the large variability of the atmosphere, it is difficult to definitely establish what is the origin of this climatic difference, but it appears that the atmospheric circulation anomaly helps sustaining the activity of this convection sites. Far from the North Atlantic, the climatic response is of small amplitude, the only significant change appearing in summer over the tropical Atlantic, where the Inter-Tropical Convergence Zone (ITCZ) shifts southward. The climate differences between the 15 Sv and 2 Sv simulations are much larger and our analyses focus on three areas: the North Atlantic and surrounding regions, the Tropics and the Indian monsoon region. We study the timing of appearance of these responses to the AMOC shutdown, which gives some clues about the mechanisms for these teleconnections. We show that the North Atlantic cooling associated with the collapse of the AMOC induces a cyclonic atmospheric circulation anomaly centered over the North Atlantic, which modulates the eastward advection of the cold anomaly over the Eurasian continent. It can explain that the cooling is not as strong over Western Europe as over the North Atlantic and the rest of the Eurasian continent. Another modification in the northern extratropical stationary waves occurs over the Eastern Pacific, explaining a warming over Northwestern America. In the Tropics, the ITCZ southward shift in this simulation appears to be strongest over the Atlantic and Eastern Pacific and results from an ajustment of the atmospheric and oceanic transports. Finally, the Indian monsoon weakening also appears to be connected to the tropospheric cooling over Eurasia.