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We propose a novel type of planar-to-spatial deployable structures that we call elastic geodesic grids. Our approach aims at the approximation of doubly curved free-form surfaces with spatial grids of bent lamella which can be deployed from a planar configuration using a simple kinematic mechanism. Such grids result in easy to fabricate and easy to deploy spatial structures which combine physics and aesthetics. Based on theoretical considerations from differential geometry, we propose an entire framework for the design, generation, physical simulation, and fabrication of elastic geodesic grids. We also present several results with varying complexity as well as fabricated prototypical examples. Moreover, we provide an empirical proof of our approach by comparing computed results to fabricated models. Our method is intended as a design and form-finding tool in architecture and other creative disciplines and it gives the designer a fast and easy way for the creation of complex physical structures.