In the past decade hundreds of galaxy candidates have been identified in the Epoch of Reionization (EoR), selected from their rest-frame UV light. Only a handful of these sources, however, have spectroscopic redshift determinations and we have limited understanding of their properties. ALMA is currently transforming this field by identifying massive ISM reservoirs at z>6.5 from bright [CII] line emission. I will describe how we obtained the first spectroscopic confirmations of galaxies in the EoR with ALMA, where Lya often remains undetected due to the intervening neutral IGM, and how ALMA is now rapidly becoming a 'redshift machine' for the distant Universe. With the spectroscopic redshifts in hand, we are in a position to start probing the physical properties of these sources, including their fine structure line ratios, dust content and temperature, structural properties and low-resolution kinematics. A picture is starting to emerge of complex structures within these galaxies with distinct dusty and UV-bright components which exhibit different ISM properties. Moreover, we find evidence that some of these galaxies are already rotation-dominated disk-like sources, just 800 million years after the Big Bang.