We examine the difference between several notions of curvature homogeneity and show that the notions introduced by Kowalski and Vanzurova are genuine generalizations of the ordinary notion of k-curvature homogeneity. The homothety group plays an essential role in the analysis. We give a complete classification of homothety homogeneous manifolds which are not homogeneous and which are not VSI and show that such manifolds are cohomogeneity one. We also give a complete description of the local geometry if the homothety character defines a split extension. Joint work with E. Garcia-Rio and P. Gilkey.