This thesis is concerned with the weighted L² boundedness of a family of convolution operators on the line with oscillating kernels. It is proved that these convolution operators are bounded from L²(w) to L²(W) where the Borel measures w and W are in a correspondence given by a maximal function and there is a sense in which this maximal function is the best possible. It is also shown that a one-weighted L² estimate holds for a family of convolution operators with radial oscillating kernels on n-dimensional space.