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We show that continuous filling or wedge-wetting transitions are possible in 3D wedge-geometries made from (angled) substrates exhibiting first-order wetting transitions and develop a comprehensive fluctuation theory yielding a complete classification of the critical behaviour. Our fluctuation theory is based on the derivation of a Ginzburg criterion for filling and also an exact transfer-matrix analysis of a novel effective Hamiltonian which we propose as a model for wedge fluctuation effects. The influence of interfacial fluctuations is shown to be very strong and, in particular, leads to a remarkable universal divergence of the interfacial roughness ξ ⊥ ∼ (TF − T ) −1/4 on approaching the filling temperature TF , valid for all possible types of intermolecular forces.PACS numbers: 68.45. Gd, 68.35.Rh, There are two reasons why it is extremely difficult to observe interfacial fluctuation effects at continuous (critical) wetting transitions in the laboratory [1]. Firstly, critical wetting is a rather rare phenomenon for which no examples are known for solid-liquid interfaces and only a limited number for fluid-fluid interfaces [2,3]. Secondly, the influence of interfacial fluctuations in three dimensions (d = 3) is believed to be rather small [1]. For example, for systems with long-ranged forces, the divergence of the wetting layer thickness ℓ on approaching the wetting temperature T w is mean-field-like, ℓ ∼ (T w − T ) −1 , and the only predicted effect of fluctuations is to induce an extremely weak divergence of the width (roughness) ξ ⊥ of the unbinding interface: ξ ⊥ ∼ − ln(T w − T ). Nonclassical critical exponents and an appreciable interfacial width are only predicted for systems with strictly shortranged forces [4], but even here the size of the asymptotic critical regime is very small and beyond the reach of current experimental and simulation methods [3,5,6].The purpose of the present article is to show that these problems do not arise for continuous (critical) filling or wedge-wetting transitions [7][8][9] occurring for fluid adsorption in three-dimensional wedges. First, we show, contrary to previous statements in the literature [8], that critical filling can occur in systems made from walls that exhibit first-order wetting transitions. Consequently, the observation of critical filling transitions is a realistic experimental prospect. Second, we argue that interfacial fluctuations have a strong influence on the character of the filling transition and, in particular, the interfacial roughness of the unbinding interface, which is shown to diverge with a universal critical exponent. The fluctuation theory we develop is based on the derivation of a Ginzburg criterion for the self-consistency of mean-field (MF) theory and also an exact transfer matrix analysis of a novel interfacial Hamiltonian model for wedge wetting which we introduce to account for the highly anisotropic soft-mode fluctuations. This model leads to a complete classification of the critical behaviour in d = 3 and predicts some remarkable...