We review recent work on scaling properties in turbulence and dynamically roughening interfaces. Although these two phenomena seem quite different, they appear to share many common features like scaling invariances, Galilean invariance, intermittency and multiscaling. In order to investigate very high Reynolds number we approximate the Navier-Stokes equations by a shell model, where wave vectors only assign to a discrete set. From numerical simulations we observe corrections to Kolmogorov theory due to intermittency effects of the dissipation structure. The temporal development of the studied interfaces is governed by punctuated dynamics. This kind of dynamics drives the evolving interfaces (or fronts) to a critical state with temporal multiscaling of the interface profile due to intermittent activity characterized by avalanches which connect regions of recent local activity. This activity pattern exhibits non-trivial power law correlations in space and time. We present numerical results in both (1 + 1) and (2 + 1) dimensions and generalize a theory for the critical exponents to all dimensions.
