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Displaying line data is important in many visualization applications, and especially in the context of interactive geographical and cartographic visualization. When rendering linear features as roads, rivers or movement data on zoomable maps, the challenge is to display the data in an appropriate level of detail. A too detailed representation results in slow rendering and cluttered maps, while a too coarse representation might miss important data aspects. In this paper, we propose the gradual line simplification (GLS) problem, which aims to compute a fine-grained succession of consistent simplifications of a given input polyline with certain quality guarantees. The core concept of gradual simplification is to iteratively remove points from the polyline to obtain increasingly coarser representations. We devise two objective functions to guide this simplification process and present dynamic programs that compute the optimal solutions in 𝒪(n³) for an input line with n points. For practical application to large inputs, we also devise significantly faster greedy algorithms that provide constant factor guarantees for both problem variants at once. In an extensive experimental study on real-world data, we demonstrate that our algorithms are capable of producing simplification sequences of high quality within milliseconds on polylines consisting of over half a million points.