It is the aim of this paper to give a unified setting to and report on a new area of research in (universal) algebra: the theory of equationally compact universal algebras. We attempt in this study to present in a concise fashion all essential results known on the subject matter as far as they have bearing on algebras rather than relational systems. More precisely: We take into account all relevant results in mathematical journals, a few unpublished results if they aid in illuminating the scope of some of the published results and this author's own research. Proofs will not be given in detail if they have appeared in print. Exception to this rule will be made for two reasons only: (1) if new ideas intrude upon a known result, (2) if the readability of this article seems jeopardized by too much omission. The paper will be concluded with a list of open problems.
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