  The idea of combining different two-dimensional (2D) crystals in van der
Waals heterostructures (vdWHs) has led to a new paradigm for band structure
engineering with atomic precision. Due to the weak interlayer couplings, the
band structures of the individual 2D crystals are largely preserved upon
formation of the heterostructure. However, regardless of the details of the
interlayer hybridisation, the size of the 2D crystal band gaps are always
reduced due to the enhanced dielectric screening provided by the surrounding
layers. The effect can be on the order of electron volts, but its precise
magnitude is non-trivial to predict because of the non-local nature of the
screening in quasi-2D materials, and it is not captured by effective
single-particle methods such as density functional theory. Here we present an
efficient and general method for calculating the band gap renormalization of a
2D material embedded in an arbitrary vdWH. The method evaluates the change in
the GW self-energy of the 2D material from the change in the screened Coulomb
interaction. The latter is obtained using the quantum-electrostatic
heterostructure (QEH) model. We benchmark the G$\Delta$W method against full
first-principles GW calculations and use it to unravel the importance of
screening-induced band structure renormalisation in various vdWHs. A main
result is the observation that the size of the band gap reduction of a given 2D
material when inserted into a heterostructure scales inversely with the
polarisability of the 2D material. Our work demonstrates that dielectric
engineering \emph{via} van der Waals heterostructuring represents a promising
strategy for tailoring the band structure of 2D materials.
