  In this survey we provide an overview of some recent developments in the
construction of moduli spaces using stack-theoretic techniques. We will also
explain the analogue of Harder-Narasimhan stratifications for general stacks,
known as $\Theta$-stratifications. As an application of the ideas exposed here,
we address the moduli problem of principal bundles over higher dimensional
projective varieties, as well as its different compactifications by the
so-called principal $\rho$-sheaves. We construct a stratification by
instability types whose lower strata admits a proper good moduli space of
``Gieseker semistable" objects and a new Gieseker-type Harder-Narasimhan
filtration for these objects. Detailed proofs of the latter results will appear
elsewhere.
