Let ${gamma}$ : Ilongrightarrow R2 be a sufficiently smooth curve and $sigma$${gamma}$ be the affine arclength measure supported on ${gamma}$. In this paper, we study the Lp - improving properties of the convolution operators T$sigma$${gamma}$ associated with $sigma$${gamma}$ for various curves ${gamma}$. Optimal results are obtained for all finite type plane curves and homogeneous curves (possibly blowing up at the origin). As an attempt to extend this result to infinitely flat curves we give and example of a family of flat curves whose affine arclength measure has same Lp-improvement property. All of these results will be based on uniform estimates of damping oscillatory integrals.