We characterize metrizable separable spaces X such that almost every, in the sense of Baire category, embedding h of X into the Hilbert cube Iω provides a compact extension h(X) such that the remainder h(X) \h(X) has certain dimensional property (for instance, is ^-dimensional, countable-dimensional or "metrically weakly infinitedimensional")* We obtain a characterizati on of metrizable separable spaces which have large transfinite dimension by means of compactifications. Two examples related to the results mentioned above are constructed. 1. Introduction. Consider the following two classes of separable metrizable spaces: the class (Pn) of spaces X with dimX Iω provides a compact extension h(X) with dimh(X) < n, the existence of a compactification Ϋ of Y whose remainder Ϋ\ Y is in the class (P), for any of the two (P) we consider, is not enough to guarantee that almost every embedding h: Y -+ Iω provides such a compactification h(Y) for h(Y). In the case of (Pn) the characterization is simple: the class consists exactly of spaces which are unions of a compact set and an ^-dimensional set (§3). To give a characterization for (Pω), we introduce a somewhat weaker property than the weak infinite-dimensionality (in the sense of Smirnov). In particular, spaces having large transfinite dimension have this property, which yields a characterizati on of spaces with trlnd, analogous to that given by Hurewicz for small transfinite dimension (§4).