Spatial approximations have been traditionally used in spatial databases to accelerate the processing of complex geometric operations. However, approximations are typically only used in a first filtering step to determine a set of candidate spatial objects that may fulfill the query condition. To provide accurate results, the exact geometries of the candidate objects are tested against the query condition, which is typically an expensive operation. Nevertheless, many emerging applications (e.g., visualization tools) require interactive responses, while only needing approximate results. Besides, real-world geospatial data is inherently imprecise, which makes exact data processing unnecessary. Given the uncertainty associated with spatial data and the relaxed precision requirements of many applications, this vision paper advocates for approximate spatial data processing techniques that omit exact geometric tests and provide final answers solely on the basis of (fine-grained) approximations. Thanks to recent hardware advances, this vision can be realized today. Furthermore, our approximate techniques employ a distance-based error bound, i.e., a bound on the maximum spatial distance between false (or missing) and exact results which is crucial for meaningful analyses. This bound allows to control the precision of the approximation and trade accuracy for performance.