Rudin's classic treatise "Function Theory in Polydiscs" (1969) studies complex analysis on polydiscs starting from the following question: How much of our extremely detailed knowledge about holomorphic functions in the unit disc can be carried over to an analogous situation in several variables, namely to polydiscs?
A different perspective comes from the work of H. Bohr and Bohnenblust--Hille several decades earlier; here the main issue is function theory in the infinite-dimensional polydisc or, more precisely, estimates in finite-dimensional polydiscs that are independent of the dimension.