In this paper the author shows that it is possible to approximate the Riemann zeta-function by functions meromorphic on \(\mathbb C\) which satisfy the functional equation of the Riemann zeta-function and do not satisfy the analogue of the Riemann hypothesis as well as to approximate the Riemann zeta-function by functions meromorphic on \(\mathbb C\) which satisfy the functional equation of the Riemann zeta-function and do satisfy the analogue of the Riemann hypothesis. More precisely he proves the following two theorems. Theorem 1. Let \(\varepsilon_n\) be a sequence of positive continuous functions decreasing to zero. There is an increasing sequence \(\{X_n\}\) of closed sets, whose union is \(\mathbb C,\) such that \(X_n\) contains closed disc \(|z-1/2|\leq r_n, \;\;( r_n \to\infty) ,\) as well as the real and the critical axis. There is a sequence \(\{\zeta_n\}\) of functions meromorphic on \(\mathbb C\) which satisfy the functional equation of the Riemann zeta-function, have a unique pole at \(z=1\) with residue 1, take real values on the real axis, have zeros at all of the trivial zeros of \(\zeta,\) as well as at the zeros of \(\zeta\) lying on the critical axis, but fail to satisfy the Riemann hypothesis in that they have other zeros in the fundamental strip \(\{z: 0<\text{Re}\, z<1\}.\) Moreover, they converge to \(\zeta\) everywhere as follows \( |\zeta_n(z)-\zeta(z)|<\varepsilon_n(z)\) for \(z \in X_n\). Theorem 2. Let \(A\) be the set of non-trivial zeros of \(\zeta\) off the critical axis and let \(\epsilon_n\) be a sequence of positive numbers strictly decreasing to zero. There is an increasing sequence \(\{X_n\}\) of closed sets, whose union is \({\mathbb C}\backslash A,\) and a sequence \(\{\zeta_n\}\) of functions meromorphic on \(\mathbb C\) which satisfy the functional equation of the Riemann zeta-function, have a unique pole at \(z=1\) with residue 1, take real values on the real axis, have zeros at all of the trivial zeros of \(\zeta,\) as well as at the zeros of \(\zeta\) lying on the critical axis, satisfy the Riemann hypothesis in that they have no other zeros in the fundamental strip \(\{z: 0<\text{Re}\, z<1\}.\) Moreover, they converge to \(\zeta\) everywhere as follows \( |\zeta_n(z)-\zeta(z)|<\varepsilon_n|\zeta(z)|\) for \(z \in X_n,\) and \(|\zeta_n(a)|< \epsilon_n\) for \(a \in A\). The paper contains a profound survey of results on this topic, as well as the other results relating to the approximation of meromorphic functions by translates of the Riemann zeta-function, linear combinations of translates of the Riemann zeta-function, Taylor polynomials of translates of the Riemann zeta-function.