An analysis of covariance model where the covariate effect is assumed only to be smooth is considered. The possibility of different shapes of covariate effect in different groups is also allowed and tests of equality and of parallelism across groups are constructed. These are implemented using Gasser-Maller smoothing, whose properties enable problems of bias to be avoided. Accurate moment-based approximations are available for the distribution of each test statistic. Some data on Spanish Onions are used to contrast the non-parametric approach with that of a nonlinear, but parametric, model. A simulation study is also used to explore the properties of the non-parametric tests.