One purpose of this chapter is to establish results on the rate of almost everywhere convergence of approximation processes of convolution type in Lp(Rn),where instead of a particular rate (like t μ, μ > 0, t → 0+), fractional moduli of smoothness are employed. An essential tool is a modified K-functional. Away from saturation orders these results are nearly optimal. A second purpose is to illustrate that the methods applied also work in other settings which feature a convolution/multiplier structure.