The problem whether every topological space $X$ has a compactification $Y$ such that every continuous mapping $f$ from $X$ into a compact space $Z$ has a continuous extension from $Y$ into $Z$ is answered in the negative. For some spaces $X$ such compactifications exist.
[AR] Adámek J., Rosický J.: On injectivity classes in locally presented categories. preprint (1991), (to appear in Trans. Amer. Math. Soc).
[Če] Čech E.: Topological spaces
. (revised ed. by Z.Frolík, M.Katětov) (Academia Prague 1966). MR 0211373
[DW] Dow A., Watson S.: Universal spaces
. preprint (December 1990). MR 1056167
[En] Engelking R.: Topological spaces. Heldermann Verlag Berlin (1990).
[GH] Giuli E., Hušek M.: A diagonal theorem for epireflective subcategories of Top and cowellpoweredness
. Ann. di Matem. Pura et Appl. 145 (1986), 337-346. MR 0886716
[GS] Giuli E., Simon P.: On spaces in which every bounded subset is Hausdorff
. Topology and its Appl. 37 (1990), 267-274. MR 1082937
| Zbl 0719.54009
[Ha] Harris D.: The Wallman compactification as a functor
. Gen. Top. and its Appl. 1 (1971), 273-281. MR 0292034
| Zbl 0224.54010
[He] Herrlich H.: Almost reflective subcategories of Top
. preprint (1991) (to appear in Topology and its Appl.). MR 1208677
[Hu$_1$] Hušek M.: Categorial connections between generalized proximity spaces and compactifications
. Contributions to extension theory of topological structures, Proc. Conf. Berlin 1967 (Academia Berlin 1969), 127-132. MR 0248730
[Hu$_2$] Hušek M.: Remarks on reflections
. Comment. Math. Univ. Carolinae 7 (1966), 249-259. MR 0202800