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The problem of the interplay between normal and anomalous scaling inturbulent systems stirred by a random forcing with a power law spectrum isaddressed. We consider both linear and nonlinear systems. As for the linearcase, we study passive scalars advected by a 2d velocity field in the inversecascade regime. For the nonlinear case, we review a recent investigation of 3dNavier-Stokes turbulence, and we present new quantitative results for shellmodels of turbulence. We show that to get firm statements is necessary to reachconsiderably high resolutions due to the presence of unavoidable subleadingterms affecting all correlation functions. All findings support universality ofanomalous scaling for the small scale fluctuations.Comment: 17 Latex2e pages, with 7 eps figures include

Anomalous scaling and universality in hydrodynamic systems with power-law forcing