Let \(D\) be an integral domain with quotient field qf\((D)=K\). Recall that an element \(x \in K\) is called \(w\)-integral [respectively: pseudo-integral (or \(v\)-integral)] on \(D\) if \( xI^w \subseteq I^w\) [respectively: \(xI^v \subseteq I^v\)] for some nonzero finitely generated ideal \(I\) of \(D\). The authors denote by \(D^w\) [respectively: \(\widetilde{D}\), \(\overline{D}\), \(D^\ast\)] the set of the elements \(w\)-integral [respectively: pseudo-integral, integral, almost integral] over \(D\); \(D^w\) [respectively: \(\widetilde{D}\), \(\overline{D}\), \(D^\ast\)] is called the \(w\)-integral closure [respectively: the pseudo-integral closure, the integral closure, the complete integral closure] of \(D\). It is easy to see that \( D\subseteq \overline{D} \subseteq D ^w \subseteq \widetilde{D} \subseteq D^\ast\,. \) Clearly, if \(D\) is noetherian [respectively: strongly Mori, \ Mori] then \( \overline{D} = D ^w = \widetilde{D} = D^\ast\) [respectively: \(D ^w= \widetilde{D} = D^\ast\), \ \( \widetilde{D} =D^\ast\)]. In the present paper the authors mainly investigate the properties of the \(w\)-integral closure of an integral domain. Among the main results obtained here, we mention the following: (1) Let \(D \subseteq R\) be an extension of integral domains, \(R\) is called \(t\)-linked over \(D\) if for each nonzero finitely generated ideal \(I\) of \(D\), \(I^{-1} = D\) implies \((IR)^{-1} = R\). Then: \(R\) is \(t\)-linked over \(D\) if and only if \(R = \{f/g \mid f, g \in R[X], c(g)^v = R\} \cap\) qf\((R)\). \ As a consequence, \(D^w\) is \(t\)-linked over \(D\) and \(D^w = \{f/g \mid f, g \in \overline{D}[X], c(g)^v = \overline{D}\} \cap K\). (2) Recall that an UMT-domain \(D\) is an integral domain such that every upper to zero in \(D[X]\) is a maximal \(t\)-ideal. An integrally closed UMT-domain is a Prüfer \(v\)-multiplication domain. Then: \(D\) is an UMT-domain if and only if \(D^w\) is a Prüfer \(v\)-multiplication and the set of maximal \(t\)-ideals of \(D^w\) coincides with \(\{Q \in \) Spec\((D^w) \mid Q\cap D \) is a maximal \(t\)-ideal of \(D \}\). An interested reader may find an independent general semistar-theoretic approach to some of the problems considered in this paper in \textit{M. Fontana} and \textit{K. A. Loper} [Commun. Algebra, 31, 4775--4805 (2003; Zbl 1065.13012)] and \textit{S. El Baghdadi} and \textit{M. Fontana} [Commun. Algebra, 32, 1101--1126 (2004; Zbl 1120.13016)].