Given a fixed class \(\mathbb L\) of frames, a frame \(T\) in \(\mathbb L\) is said to be universal in this class if for every \(L\in\mathbb L\) there exists a frame homomorphism of \(T\) onto \(L\). In the paper under review, the author proves that the classes of regular frames and completely regular frames have universal elements, answering in the affirmative two of the three problems proposed at the end of \textit{T. Dube} et al. [Topology Appl. 160, No. 18, 2454-2464 (2013; Zbl 1278.06001)]. The proofs are based on the adaptation to frames of the general method of construction of universal elements, in the category of topological spaces, due to the author [Universal spaces and mappings. Amsterdam: Elsevier (2005; Zbl 1072.54001)]. Reviewer's comments: Contrary to what the author claims in the introduction, the problem of finding universal elements in the pointfree setting of frames is not in its beginnings. In fact, it has been known since [\textit{J. R. Isbell}, Math. Scand. 31, 5-32 (1972; Zbl 0246.54028)] that once there are free objects in the category of frames and they are indeed topologies, then every frame is a quotient of a spatial frame. More specifically, any frame of weight \(\leq\kappa\) is a quotient of the frame \(\mathcal O(\mathbb S^\kappa)\) of open sets of the power \(\mathbb S^\kappa\) of the Sierpiński space \(\mathbb S\). In the author's terminology, this means that \(\mathcal O(\mathbb S^\kappa)\) is universal in the class of all frames of weight \(\leq\kappa\). There are various other results of that kind in the literature on locale theory (in fact, what the author actually aims at, formulated in terms of the dual category of locales, is the finding, in any given class of locales, of a locale in which every other locale of the class is embeddable): (1) A locale is completely regular if and only if it is embeddable into a product of copies of the closed unit interval locale [\textit{P. T. Johnstone}, Stone spaces. Cambridge Studies in Advanced Mathematics 3. Cambridge: Cambridge University Press (1982; Zbl 0499.54001)]. (2) Any zero-dimensional locale is embeddable into a product of copies of the four element Boolean algebra [\textit{I. Paseka}, Cah. Topologie Géom. Différ. Catégoriques 33, No. 1, 15-20 (1992; Zbl 0771.54020)]. (3) Most notably, there is a recent paper by \textit{L. Español} et al., [Algebra Univers. 67, No. 2, 105-112 (2012; Zbl 1259.06011)] where the universality problem is treated. In particular, it contains three results (namely, the Tychonoff Embedding Theorem 5.4(2), the Urysohn Embedding Theorem 5.6 and Theorem 5.4(1)) that provide immediate positive answers to the three open problems of [\textit{T. Dube} et al., loc. cit.] mentioned above.