We show some analytical, and partly numerical, results on the effective diffusivity of tracer and inertial particles in flowing fluids. Particle diffusion is a phenomenon where the mean square displacement - after subtracting its average, which corresponds to the advective or ballistic degree of freedom - follows a power law in time. In most standard cases the exponent is 1, and the tensorial prefactor is called "eddy diffusivity" and can be found by means of a multiple-scale expansion. If under investigation are tracers, i.e. fluid particles, we present very recent findings about the role of a (short) flow correlation time and the interplay between helicity, molecular diffusion and mean streaming, in collaboration with Andrea Mazzino (Univ. Genova) and Sílvio Gama, by using functional-calculus techniques and the Furutsu-Novikov-Donsker theorem. When, on the contrary, the focus is on particles endowed with a finite relative inertia, several control parameters come into play, and the problem can be attacked by studying the interference between each physical effect. In particular, situations can exist where the aforementioned exponent differs from unity, in which case one speaks of anomalous diffusion. We also illustrate an example where the classical perturbative approach fails in determining the bounds for the appearance of anomaly.