The theory of higher order elliptic equations is full of paradoxes and open problems. Contrary to Hadamard's conjecture, the positivity of the Green function may fail, and even the Miranda-Agmon maximum principle does not necessarily hold in non-smooth domains. We will discuss these and related issues. I will present recently discovered sharp pointwise estimates on polyharmonic functions in rough domains and the extension of Wiener criterion, capacity and potential to the context of higher order PDEs. This is joint work with V. Maz'ya.