It is a basic fact that Nature behaves differently at different scales. What happens at the human scale looks rather different to the motion of planets or the dynamics of subatomic particles. Rather remarkably, understanding this seemingly elementary phenomenon led to one of the most profound paradigm shifts in Physics and to the most accurate agreement between theory and experiment in Science. All thanks to the powerful tool of the Renormalisation Group.

However, in the presence of gravity this becomes rather challenging as scales become outputs of the dynamics rather than inputs. This thesis covers several studies regarding this intersection, trying to understand the interplay between scales, quantum fields, and gravity.

We begin with an overview of renormalisation techniques to set the stage for the rest of the discussion. We then delve into the AdS/CFT correspondence, first studying a local version of the renormalisation group and testing the quantum renormalisation group proposal. After that we analyse the surprising implications conformal and supersymmetry on the field theory side have on string theoretic corrections for its gravitational dual. Finally we explore the world of finite temperature quantum field theory and the subtleties behind correctly setting initial conditions.