[Resent message] A preprint of a paper entitled "A comonadic account of behavioural covarieties of coalgebras" is available for downloading as a pdf file from www.mcs.vuw.ac.nz/~rob Rob Goldblatt ABSTRACT A class $K$ of coalgebras for an endofunctor $T:\Set\to\Set$ is a \textit{behavioural covariety} if it is closed under disjoint unions and images of bisimulation relations (hence closed under images and domains of coalgebraic morphisms, including subcoalgebras). $K$ may be thought of as the class of all coalgebras that satisfy some computationally significant property. In any logical system suitable for specifying properties of state-transition systems in the Hennessy-Milner style, each formula will define a class of models that is a behavioural variety. Assume that the forgetful functor on $T$-coalgebras has a right adjoint, providing for the construction of cofree coalgebras, and let $\G^T$ be the comonad arising from this adjunction. Then we show that behavioural covarieties $K$ are (isomorphic to) the Eilenberg-Moore categories of coalgebras for certain comonads $\G^K$ naturally associated with $\G^T$. These are called \textit{pure subcomonads} of $\G^T$, and a categorical characterization of them is given, involving a pullback condition on the naturality squares of a transformation from $\G^K$ to $\G^T$. We show that there is a bijective correspondence between behavioural covarieties of $T$-coalgebras and isomorphism classes of pure subcomonads of $\G^T$.