The inverse scattering transform (IST) as a tool to solve the initial-value problem for the focusing nonlinear Schrödinger (NLS) equation with non-zero boundary values \documentclass[12pt]{minimal}\begin{document}$q_{l/r}(t)\equiv A_{l/r} e^{-2iA_{l/r}^2t+i\theta _{l/r}}$\end{document}ql/r(t)≡Al/re−2iAl/r2t+iθl/r as x → ∓∞ is presented in the fully asymmetric case for both asymptotic amplitudes and phases, i.e., with Al ≠ Ar and θl ≠ θr. The direct problem is shown to be well-defined for NLS solutions q(x, t) such that \documentclass[12pt]{minimal}\begin{document}$\left(q(x,t)-q_{l/r}(t)\right)\in L^{1,1}(\mathbb {R^\mp })$\end{document}q(x,t)−ql/r(t)∈L1,1(R∓) with respect to x for all t ⩾ 0, and the corresponding analyticity properties of eigenfunctions and scattering data are established. The inverse scattering problem is formulated both via (left and right) Marchenko integral equations, and as a Riemann-Hilbert problem on a single sheet of the scattering variables \documentclass[12pt]{minimal}\begin{document}$\lambda _{l/r}=\sqrt{k^2+A^2_{l/r}}$\end{document}λl/r=k2+Al/r2, where k is the usual complex scattering parameter in the IST. The time evolution of the scattering coefficients is then derived, showing that, unlike the case of solutions with equal amplitudes as x → ±∞, here both reflection and transmission coefficients have a nontrivial (although explicit) time dependence. The results presented in this paper will be instrumental for the investigation of the long-time asymptotic behavior of fairly general NLS solutions with nontrivial boundary conditions via the nonlinear steepest descent method on the Riemann-Hilbert problem, or via matched asymptotic expansions on the Marchenko integral equations.