Algebraically, principal components can be defined as the eigenvalues and eigenvectors of a covariance or correlation matrix, but they are statistically meaningful as successive projections of the multivariate data in the direction of maximal variability. An attractive alternative in robust principal component analysis is to replace the classical variability measure, i.e. variance, by a robust dispersion measure. This projection‐pursuit approach was first proposed in Li & Chen (1985) as a method of constructing a robust scatter matrix. Recent unpublished work of C. Croux and A. Ruiz‐Gazen provided the influence functions of the resulting principal components. The present paper focuses on the asymptotic distributions of robust principal components. In particular, we obtain the asymptotic normality of the principal components that maximise a robust dispersion measure. We also explain the need to use a dispersion functional with a continuous influence function.