  We analyze the one-dimensional periodic Kardar-Parisi-Zhang equation in the
language of paracontrolled distributions, giving an alternative viewpoint on
the seminal results of Hairer.
  Apart from deriving a basic existence and uniqueness result for
paracontrolled solutions to the KPZ equation we perform a thorough study of
some related problems. We rigorously prove the links between KPZ equation,
stochastic Burgers equation, and (linear) stochastic heat equation and also the
existence of solutions starting from quite irregular initial conditions. We
also build a partial link between energy solutions as introduced by Goncalves
and Jara and paracontrolled solutions.
  Interpreting the KPZ equation as the value function of an optimal control
problem, we give a pathwise proof for the global existence of solutions and
thus for the strict positivity of solutions to the stochastic heat equation.
  Moreover, we study Sasamoto-Spohn type discretizations of the stochastic
Burgers equation and show that their limit solves the continuous Burgers
equation possibly with an additional linear transport term. As an application,
we give a proof of the invariance of the white noise for the stochastic Burgers
equation which does not rely on the Cole-Hopf transform.
