Passive scalars advected by irregular velocity fields provide a model problem for turbulent behaviour, namely the cascade of scalar variance to small scales. In this talk I will focus on the vanishing diffusivity regime for transport?diffusion: when the advecting velocity is rough, the inviscid transport equation may lose classical well-posedness and the expected conservation of the energy. Although any fixed positive diffusivity restores a unique, well-posed evolution, the limit as diffusivity tends to zero can exhibit turbulent phenomena such as anomalous dissipation and failure of the diffusive regularization to select a unique inviscid solution.