Consider a smooth quasi-projective variety \(U/\mathbb C\) with its Bloch’s higher Chow groups and the mixed Hodge structure on its cohomology. A. A. Beilinson [Applications of algebraic \(K\)-theory to algebraic geometry and number theory. Contemp. Math. 55, 35–68 (1986; Zbl 0621.14011)] conjectured that the cycle class map \(\mathrm{cl}_{r,m} : \mathrm{CH}^{r}(U,m)\otimes \mathbb Q \to \hom_{\mathrm{MHS}}(\mathbb Q(0),H^{2r-m}(U, \mathbb Q(r)))\) should be surjective. In his Habilitationsschrift [Mixed motives and algebraic K-theory. Berlin etc.: Springer-Verlag (1990; Zbl 0691.14001)], U. Jannsen noted that it is simple to find a counterexample in the case \(m=1\), by making wise use of the snake lemma. The main aim of the authors of the present paper is to discuss at length conditions and obstructions to the surjectivity statement. Their method can be described as a homological tour de force, based on the study and comparison of the weight filtered spectral sequences, which appear in the construction of \(\mathrm{CH}^{r}(U,m) \) and of \(H^{2r-m}(U,m)\), when \( U = X \setminus Y\), where \(X\) is a smooth projective variety and \(Y = Y_{1}\cup\cdots\cup Y_{n}\subset X\) a normal crossing divisor with smooth components. The special cases \(m = 1 \) and \(m = r = 2\) are presented as examplifications. There is a section, by the title amending the Beilinson-Hodge conjecture. It depends on Jannsen’s example on \(U\), the idea is to consider products \(U \times {\mathbb {C}^{*}}^{\times (m-1)}\), there everything is computable by iteration in terms of \(U\). The outcome is the observation that there are counterexamples to the surjectivity of \(\mathrm{cl}_{r,m}\) for all \(m< r \leq \dim(U)\), and then the only interesting cases of the conjecture which remain open are either \(m=0\) or \(r=m\). On the other hand the authors expect that the conjecture holds in the limit over all Zariski open \(U\) inside a fixed complex projective variety \(X\), and the statement is then the surjectivity of \(\lim (\mathrm{cl}_{r,m}): \mathrm{CH}^{r}(\mathrm{Spec}(\mathbb C(X)),m) \otimes \mathbb Q \to \hom_{\mathrm{MHS}}(\mathbb Q(0),H^{2r-m}(\mathrm{Spec}(\mathbb C(X)), \mathbb Q(r)))\). For the special case \(r=m\) an apparently finer conjecture is proposed, to the effect that the preceding map is surjective even with integer coefficients. An important motivation for such a proposal is the theorem that any torsion class in \(H^n(X,\mathbb Z)\) is supported in codimension one. In [S. Bloch, Lectures on algebraic cycles. Durham, North Carolina: Duke University, Mathematics Department. Not consecutively paged (1980; Zbl 0436.14003)], see p. 5.13, Bloch made the deep observation that what was to become a conjecture of his and Kato implies the statement on torsion, and now that conjecture is a theorem. The authors prove that it then follows from this that it amounts to the same thing to ask for surjectivity of \(\lim(\mathrm{cl}_{m,m})\) either with \(\mathbb Z\) coefficients or with \(\mathbb Q\) coefficients. In a rather sketchy final section there is a proposal of construction of a family of open surfaces \(U_t\) having interesting MHS on \(H^{2}(U_t, \mathbb Q(2))\), the motivation being that it might be possible to test the Hodge-Beilinson conjecture for them. The reader is not told much as to what the MHS turns out to be in the cases presented, may be that such information can be extrapolated from J. A. Carlson’s theorem K in [Compos. Math. 56, 271–314 (1985; Zbl 0629.14027)].