Computing the viscosity solution of eikonal equations remains a fundamental challenge in scientific computing, with applications spanning computer graphics, path planning, and wave propagation. This talk surveys comprehensive research trajectory from classical finite volume methods to modern scientific machine learning approaches for solving eikonal equations. We begin with cell-centered finite volume methods on polyhedral meshes, where the Soner boundary condition shows essential for numerical convergence to the viscosity solution. The initial work establishes second order accuracy for the time-relaxed eikonal equation, while subsequent Laplacian regularization dramatically reduces computational costs by solving a steady-state formulation instead of time-dependent iterations. These classical approaches provide robust solutions but face generation of meshes. Transitioning to neural network methods, we address the notorious challenge that conventional Physics-Informed Neural Networks often converge to spurious solutions rather than the unique viscosity solution. The Neural Augmented Lagrangian Method recasts the problem as constrained optimization, maximizing solution volume subject to 1-Lipschitz gradient constraints while incorporating Soner boundary conditions. This framework rigorously enforces the variational characterization of viscosity solutions. For anisotropic problems, we introduce a viscosity reduction variational approach with variable splitting and normalized neural representations, overcoming nonlinearity and singular perturbation challenges inherent in vanishing viscosity methods. Most recently, the stochastic vanishing viscosity approach with the Derivative Free Loss Method captures directional transport along gradient trajectories, aligning with physical propagation mechanisms rather than enforcing pointwise partial differential equation residuals. The mentioned methods demonstrate the following properties: mesh-free formulation, use of various viscosity scales, and consistent convergence across diverse source configurations and spatially varying speeds.