How to weight gravity?

In spite of General Relativity Theory being around for over a century, we still do not have a satisfactory understanding of the concept of energy of the gravitational field.
Although one can assign a value of total energy to some gravitating systems, the very nature of the theory makes it impossible to define local energy density of gravity.
As a way around this, people try to define energy for finite, but extended regions of spacetimes. This is known as the problem of quasi-local mass.
I will discuss some of the difficulties that make the search for a viable definition of quasi-local energy so challenging and show how the  approximated theory of linearized gravity may offer clues to dealing with them.