In this paper we address absolute camera pose estimation. An efficient (and standard) way to solve this problem, is to use sparse keypoint correspondences. In many cases point features are not available, or are unstable over time and viewing conditions. We propose a framework based on silhouettes of quadric surfaces, with special emphasis on cylinders. We provide mathematical analysis of the problem of projected cylinders in particular, but also general quadrics. We develop a number of minimal solvers for estimating camera pose from silhouette lines of cylinders, given different calibration and cylinder properties. These solvers can be used efficiently in bootstrapping robust estimation schemes, such as RANSAC. Note that even though we have lines as image features, this is a different case than line based pose estimation, since we do not have 2D-line to 3D-line correspondences. We perform synthetic accuracy and robustness tests and evaluate on a number of real case scenarios.