In this talk, I will consider quasilinear parabolic PDEs subject to stochastic or rough perturbation and explain how various assumptions on coefficients and roughness of the noise naturally ask for different notions of solution with different regularity properties and different techniques of the proofs. On the one hand, the problems under consideration will be stochastic second order parabolic PDEs with noise smooth in space, either with a possible degeneracy in the leading order operator, where only low regularity holds true, or under the uniform ellipticity assumption, where arbitrarily high regularity can be proved under suitable assumptions on the coefficients. On the other hand, I will discuss a rough pathwise approach towards these problems based on tools from paracontrolled calculus.