In this article, we present four cases of minimal solutions for two-view relative pose estimation by exploiting the affine transformation between feature points, and we demonstrate efficient solvers for these cases. It is shown that under the planar motion assumption or with knowledge of a vertical direction, a single affine correspondence is sufficient to recover the relative camera pose. The four cases considered are two-view planar relative motion for calibrated cameras as a closed-form and least-squares solutions, a closed-form solution for unknown focal length, and the case of a known vertical direction. These algorithms can be used efficiently for outlier detection within a RANSAC loop and for initial motion estimation. All the methods are evaluated on both synthetic data and real-world datasets. The experimental results demonstrate that our methods outperform comparable state-of-the-art methods in accuracy with the benefit of a reduced number of needed RANSAC iterations. The source code is released at https://github.com/jizhaox/relative_pose_from_affine.