Summary: We develop the representation theory of shifted quantum affine algebras \(\mathcal{U}_\mu (\hat{\mathfrak{g}})\) and of their truncations, which appeared in the study of quantized K-theoretic Coulomb branches of 3d \(N=4\) SUSY quiver gauge theories. Our approach is based on novel techniques, which are new in the cases of shifted Yangians or ordinary quantum affine algebras as well: realization in terms of asymptotical subalgebras of the quantum affine algebra \(\mathcal{U}_q(\hat{\mathfrak{g}})\), induction and restriction functors to the category \(\mathcal{O}\) of representations of the Borel subalgebra \(\mathcal{U}_q(\hat{\mathfrak{b}})\) of \(\mathcal{U}_q(\hat{\mathfrak{g}})\), relations between truncations and Baxter polynomiality in quantum integrable models, and parametrization of simple modules via Langlands dual interpolation. We first introduce the category \(\mathcal{O}_\mu\) of representations of \(\mathcal{U}_\mu (\hat{\mathfrak{g}})\) and we classify its simple objects. Then we establish the existence of fusion products and we get a ring structure on the sum of the Grothendieck groups \(K_0(\mathcal{O}_\mu)\). We classify simple finite-dimensional representations of \(\mathcal{U}_\mu (\hat{\mathfrak{g}})\) and we obtain a cluster algebra structure on the Grothendieck ring of finite-dimensional representations. We prove a truncation has only a finite number of simple representations and we introduce a related partial ordering on simple modules. Eventually, we state a conjecture on the parametrization of simple modules of a non-simply-laced truncation in terms of the Langlands dual Lie algebra. We have several evidences, including a general result for simple finite-dimensional representations.