On a formal level, the Dirichlet space on the bidisc is the tensor product of two copies of the classical holomorphic Dirichlet space. Multipliers and Carleson measures for the space were recently characterized, and the results have been extended to the tri-disc, but not to higher tensor powers. Underlying all this there is a new multi-parameter potential theory which is still in its infancy, and many basic problems await an answer. The talk reports on work by several authors: Pavel Mozolyako, Karl-Mikael Perfekt, Giulia Sarfatti, Irina Holmes, Alexander Volberg, Georgios Psaromiligkos, Pavel Zorin-Kranich, and the speaker.