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Let F be a finite set of graphs. In the F-Deletion problem, we are given ann-vertex graph G and an integer k as input, and asked whether at most kvertices can be deleted from G such that the resulting graph does not contain agraph from F as a minor. F-Deletion is a generic problem and by selectingdifferent sets of forbidden minors F, one can obtain various fundamentalproblems such as Vertex Cover, Feedback Vertex Set or Treewidth t-Deletion.  In this paper we obtain a number of generic algorithmic results about PlanarF-Deletion, when F contains at least one planar graph. The highlights of ourwork are - A constant factor approximation algorithm for the optimizationversion of Planar F-Deletion. - A randomized linear time and single exponentialparameterized algorithm, that is, an algorithm running in time O(2^{O(k)} n),for the parameterized version of Planar F-deletion where all graphs in F areconnected. The algorithm can be made deterministic at the cost of making thepolynomial factor in the running time n*log^2 n rather than linear. - Apolynomial kernel for parameterized Planar F-deletion  These algorithms unify, generalize, and improve a multitude of results in theliterature. Our main results have several direct applications, but also themethods we develop on the way have applicability beyond the scope of thispaper. Our results -- constant factor approximation, polynomial kernelizationand FPT algorithms -- are stringed together by a common theme of polynomialtime preprocessing.

Planar F-Deletion: Approximation, Kernelization and Optimal FPT Algorithms