The Lehmann-Goerisch theorem is a useful method for obtaining high-precision eigenvalue bounds for differential operators. However, its advantages, such as the ability to use non-uniform meshes and high-order finite element spaces, have been underappreciated for long time. This presentation will provide an overview of the Lehmann-Goerisch theorem, its application to two model eigenvalue problems (the Laplacian eigenvalue problem and the Steklov eigenvalue problem), and the relationship between Lehmann-Goerisch's theorem and Kato's and Temple's bounds.