In this paper, for an arbitrary Kac-Moody Lie algebra {\mathfrak g} and a diagram automorphism \mu of {\mathfrak g} satisfying certain natural linking conditions, we introduce and study a \mu-twisted quantum affinization algebra {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right) of {\mathfrak g}. When {\mathfrak g} is of finite type, {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right) is Drinfeld's current algebra realization of the twisted quantum affine algebra. When \mu =\mathrm {id} and {\mathfrak g} in affine type, {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right) is the quantum toroidal algebra introduced by Ginzburg, Kapranov and Vasserot. As the main results of this paper, we first prove a triangular decomposition for {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right). Second, we give a simple characterization of the affine quantum Serre relations on restricted {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right)-modules in terms of "normal order products". Third, we prove that the category of restricted {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right)-modules is a monoidal category and hence obtain a topological Hopf algebra structure on the "restricted completion" of {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right). Last, we study the classical limit of {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right) and abridge it to the quantization theory of extended affine Lie algebras. In particular, based on a classification result of Allison-Berman-Pianzola, we obtain the \hbar-deformation of all nullity 2 extended affine Lie algebras. [ABSTRACT FROM AUTHOR]