  The paper is in essence a survey of categories having $\phi$-weighted
colimits for all the weights $\phi$ in some class $\Phi$. We introduce the
class $\Phi^+$ of {\em $\Phi$-flat} weights which are those $\psi$ for which
$\psi$-colimits commute in the base $\V$ with limits having weights in $\Phi$;
and the class $\Phi^-$ of {\em $\Phi$-atomic} weights, which are those $\psi$
for which $\psi$-limits commute in the base $\V$ with colimits having weights
in $\Phi$. We show that both these classes are {\em saturated} (that is, what
was called {\em closed} in the terminology of \cite{AK88}). We prove that for
the class $\p$ of {\em all} weights, the classes $\p^+$ and $\p^-$ both
coincide with the class $\Q$ of {\em absolute} weights. For any class $\Phi$
and any category $\A$, we have the free $\Phi$-cocompletion $\Phi(\A)$ of $\A$;
and we recognize $\Q(\A)$ as the Cauchy-completion of $\A$. We study the
equivalence between ${(\Q(\A^{op}))}^{op}$ and $\Q(\A)$, which we exhibit as
the restriction of the Isbell adjunction between ${[\A,\V]}^{op}$ and
$[\A^{op},\V]$ when $\A$ is small; and we give a new Morita theorem for any
class $\Phi$ containing $\Q$. We end with the study of $\Phi$-continuous
weights and their relation to the $\Phi$-flat weights.
