In [\textit{Paul} and \textit{Maréchal}, Nouv. Ann. (2) 15, 286--287 (1876; JFM 08.0330.01)] and [\textit{H. Brocard}, Nouv. Ann. Math. (3) 4, 291 (1885)], the following problem was posed: \textit{Pour quelles valeurs du nombre entier \(x\) l'expression} \[ 1\cdot 2\cdot 3\cdot 4\cdots x +1 \] \textit{est-elle un carré parfait?} In 1913, Ramanujan posed the same problem as follows: \textit{the number \(n! + 1\) is a square for \(n = 4, 5, 7,\) find other values.} See [\textit{B. Ram}, J. Indian Math. Soc. 5, 56--59 (1913; JFM 44.0218.01)] and [\textit{Collected papers}, New York, 1962]. Therefore, the Diophantine equation \[ n! + 1= y^2 \tag{\text{Broc-Ram}} \] is called the \textit{Brocard-Ramanujan } Diophantine equation. Solving (Broc-Ram) is still an open problem, see \textbf{D25} in [\textit{R. K. Guy}, Unsolved problems in number theory. 2nd ed. New York, NY: Springer-Verlag (1994; Zbl 0805.11001)]. Contributing to the problem, \textit{A. Gérardin} [Nouv. Ann. Math. (4) 6, 222--226 (1906; JFM 37.0230.03)], assumed that the equation (Broc-Ram) has no solutions for \(7 < n < 25\). His ideas were used by \textit{H. Gupta} see [Math. Stud. 3, 71 (1935; JFM 61.1072.03)], to prove that the equation (Broc-Ram) has no solutions except the known solutions for \(n\leq 63\). In 1993, \textit{M. Overholt} [Bull. Lond. Math. Soc. 25, No. 2, 104 (1993; Zbl 0805.11030)] proved that the weak form of Szpiro's conjecture implies that the equation (Broc-Ram) has only finitely many solutions. In fact, the weak form of Szpiro's conjecture is a special case of the \(abc\)-conjecture, i.e. there exists a constant \(s\) such that if \(a, b,\) and \(c\) are positive integers satisfying \(a + b = c\) with \(\gcd(a , b) = 1\), then \[ |abc| \leq\text{rad}(abc)^s, \] where \(\text{rad}(N)\) is the product of all primes dividing \(N\) taken without repetition. In 2000, \textit{B. C. Berndt} and \textit{W. F. Galway} [Ramanujan J. 4, No. 1, 41--42 (2000; Zbl 0999.11078)], used a computational method to show that the Brocard-Ramanujan Diophantine equation has no solutions except \((n, y) = (4, 5), (5, 11), (7,71)\) for \(n\leq 10^9\). Many other papers dealt with this equation without solving it completely. In the paper under review, the authors consider many variations of the Brocard-Ramanujan equation, particularly some equations of the form \[ y^{2} = B U_n +A, \tag{\text{Dab-Ulas}} \] where \(A\) and \(B\) are fixed integers, \(U_n=f(1) f(2)\cdots f(n)\), and \(f: \mathbb{N}_{+} \rightarrow \mathbb{N}_{+}\) is an increasing function. They completely prove many results on equations of the type (Dab-Ulas). The method is elementary using the construction of linear polynomials. Sometimes, they use a computational method. Moreover, they set many open problems and conjectures.