One of the grand scientific challenges is to overcome the curse of dimensionality in the data and physical sciences. The central theme of this proposal is that we can overcome this exponential wall by adopting and developing low-rank tensor approximations of the relevant spaces. To achieve this, we will tackle central mathematical problems such as tensor factorizations, tensor completion and the construction of tropical tensor networks, delve into computational algorithms related to tensor networks and to solving multivariate polynomial equations, and explore far-reaching applications of tensors in the fields of quantum many-body physics, multilinear system modelling and machine learning.