A theory is developed of the intricately fingered patterns of flux domains observed in the intermediate state of thin type-I superconductors. The patterns are shown to arise from the competition between the long-range Biot-Savart interactions of the Meissner currents encircling each region and the superconductor-normal surface energy. The energy of a set of such domains is expressed as a nonlocal functional of the positions of their boundaries, and a simple gradient flow in configuration space yields branched flux domains qualitatively like those seen in experiment. Connections with pattern formation in amphiphilic monolayers and magnetic fluids are emphasized. [S0031-9007(96)00250-5] When a thin film of a type-I superconductor is placed in a magnetic field normal to the sample, the large demagnetizing effects associated with the film geometry preclude the establishment of the Meissner phase (with magnetic induction B › 0). The sample instead accommodates the field by breaking up into a large number of superconducting (B › 0) and normal (B fi 0) regions, usually forming very intricate patterns [1]. As has been understood since Landau’s pioneering work [2] these structures arise from the competition between the magnetic field energy of the domains and the surface energy between the superconducting and normal regions. All existing theories of these patterns [2,3] have explored this competition with variational calculations that assume regular geometries of the flux domains. The hypothesized parallel stripes, ordered arrays of circles, etc. are rarely seen, the norm instead being the disordered patterns well documented in experiments [1,4]. Moreover, the temperature and magnetic field history of the sample strongly influence the patterns, suggesting that they are likely not in a global energetic minimum. Recent work has emphasized that diffusion of magnetic flux in the normal phase can influence the domain morphology [5]. Asymptotic methods applied to the time-dependent Ginzburg-Landau model [6] yield a free-boundary dynamics of superconductor-normal (S-N) interfaces nearly identical to that for the growth of a solid into a supercooled liquid, where the interface motion is unstable (e.g., forming dendrites). By analogy it was suggested, and confirmed by numerical studies [5], that the growth of the superconducting phase into the supercooled normal phase should be dynamically unstable, leading to highly ramified domain shapes. While such diffusive instabilities may play a role in the pattern formation in the intermediate state, these studies are not directly applicable as they have all ignored demagnetizing effects. Here we ignore entirely any diffusional instabilities and focus instead on the role of demagnetizing fields in producing the observed patterns. This is done in a completely general way, without imposing a predetermined ordered flux structure. We ask the basic question: What is the energy of a thin multiply connected superconducting domain, the normal regions of which are threaded with a magnetic field? A central issue is whether the interactions between the Meissner currents flowing along the S-N interfaces within the film are screened by the superconducting regions. Pearl [7] made the important observation that, unlike in bulk, vortices in a thin film interact with an unscreened potential V sr d, 1 y rfor large separations r, while for small r, V sr d, lnsLyrd, with L an appropriate cutoff. The lack of screening reflects the dominant role of the electromagnetic fields in the vacuum above and below the film. This suggests the simple model developed here: domains bounded by current loops interacting as in free space, endowed with line tension, and subject to the constraint of constant total magnetic flux through the sample. When applied to the stripe phase our model predicts equilibrium lengths close to those found by Landau [2] and seen in experiment [1], suggesting that it captures the essential physics. More importantly, having formulated the model for arbitrary domain shapes, we can address the origin of the disordered patterns which are so prevalent. A simplified dynamical model for the evolution of domain boundaries is used to show that the long-range interactions destabilize flux domains of regular shape, producing branched, fingered structures as seen in experiment. Apart from the global flux constraint, this model is equivalent to one for domains of magnetic fluids in Hele-Shaw flow [8], which exhibit patterns like the intermediate state [9], with history dependence like that noted earlier. Thin magnetic films [10] and monolayers of dipolar molecules [11] exhibit similar behavior, and are described by such models through the underlying correspondence between electric and magnetic dipolar phenomena [12]. This model is also very similar to a reaction-diffusion system [13] in which chemical fronts move in response to line tension and a nonlocal coupling, producing labyrinthine patterns seen in experiment [14].