  An action of a topological semigroup S on X is compactifiable if this action
is a restriction of a jointly continuous action of S on a Hausdorff compact
space Y. A topological semigroup S is compactifiable if the left action of S on
itself is compactifiable. It is well known that every Hausdorff topological
group is compactifiable. This result cannot be extended to the class of
Tychonoff topological monoids. At the same time, several natural constructions
lead to compactifiable semigroups and actions. We prove that the semigroup
C(K,K) of all continuous selfmaps on the Hilbert cube K is a universal second
countable compactifiable semigroup (semigroup version of Uspenskij's theorem).
Moreover, the Hilbert cube K under the action of C(K,K) is universal in the
realm of all compactifiable S-flows X with compactifiable S where both X and S
are second countable. We strengthen some related results of Kocak & Strauss and
Ferry & Strauss about Samuel compactifications of semigroups. Some results
concern compactifications with separately continuous actions,
LMC-compactifications and LMC-functions introduced by Mitchell.
