The paper deals with robust ANCOVA when there are one or two covariates. Let Mj (Y |X) = β0j + β1j X1 + β2j X2 be some conditional measure of location associated with the random variable Y , given X, where β0j , β1j and β2j are unknown parameters. A basic goal is testing the hypothesis H0: M1(Y |X) = M2(Y |X). A classic ANCOVA method is aimed at addressing this goal, but it is well known that violating the underlying assumptions (normality, parallel regression lines and two types of homoscedasticity) create serious practical concerns. Methods are available for dealing with heteroscedasticity and nonnormality, and there are well-known techniques for controlling the probability of one or more Type I errors. But some practical concerns remain, which are reviewed in the paper. An alternative approach is suggested and found to have a distinct power advantage.