This article is a continuation of an earlier one by the same authors [Lect. Notes Math. 883, 1-129 (1981; Zbl 0497.03025)]. It contains various sufficient conditions for representability of cylindric algebras; also three methods for constructing non-representable algebras are described. In particular, one can find one more proof of the well-known Tarski's theorem that every \(Lf_{\alpha}\) with \(\alpha\geq \omega\) is representable; the present proof is a version of Henkin's proof of the completeness theorem for first order logic. Reviewer's remark: The text of the article, almost word for word, has also been included in Sect. 3.2 (items 3.2.1-3.2.72) of the authors' monograph [Cylindric algebras, Part II (1985; Zbl 0576.03043)].