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A rigorous theoretical investigation is made to study the existence, the formation, and the basic properties of the dust-electron-acoustic (DEA) solitary waves (SWs) and double layers (DLs) in a dusty nonthermal plasma containing nonthermal ions, cold electrons, Maxwellian hot electrons, and arbitrarily (positively or negatively) charged stationary dust, by employing the reductive perturbation technique. The standard Korteweg–de Vries (K–dV), modified K–dV (mK–dV), and standard Gardner (SG) equations describing the nonlinear propagation of the DEA waves are derived, and SW solutions of the K–dV and mK–dV equations, and SW and DL solutions of the SG equation are numerically analyzed. The parametric regimes for the existence of K–dV, mK–dV, and Gardner solitons (GSs; associated with either positive or negative potential), and DLs (associated with positive potential only) are obtained. The basic properties of some interesting nonlinear structures such as DEA SWs and DLs are found to be significantly modified by the effects of the polarity of charged dust and nonthermal ions. The basic differences among GSs, K–dV solitons, and mK–dV solitons, which are identified by this investigation, are pinpointed. The implications of our results in both space and laboratory dusty plasmas are briefly discussed.