Traditional numerical methods, supported by rigorous mathematical theory, high accuracy, and physical conservation, form the cornerstone of modern scientific computing. Nevertheless, they face practical and methodological challenges, including mesh generation for complex geometries, limited ability to capture global structures, repeated reformulation under varying geometries or boundary conditions, the curse of dimensionality, and difficulties in integrating data and uncertainty. Neural-network-based methods have recently emerged as a promising alternative due to their strong expressive power, but conventional training-based approaches remain constrained by nonlinear and non-convex optimization, which limits both accuracy and efficiency. To overcome these difficulties, we propose a family of Randomized Neural Network (RaNN) methods that integrate the mathematical rigor of classical numerical formulations with the flexibility of neural representations. The framework encompasses RaNN-Petrov-Galerkin (RaNN-PG), local RaNN-DG (LRaNN-DG), LRaNN-HDPG, and LRaNN-finite difference methods. We further introduce an Adaptive-Growth RaNN (AG-RaNN) strategy that uses prior and posterior information to identify informative features, adapt random parameter distributions, and dynamically refine the network architecture, substantially improving approximation accuracy. We also investigate RaNN-based acceleration of operator learning for parameterized PDEs. Numerical experiments show that RaNN methods are mesh-free, structure-preserving, and highly expressive, achieving high accuracy with relatively few degrees of freedom and extending naturally to high-dimensional and time-dependent problems. These results highlight RaNN as a promising direction for unifying traditional numerical methods with modern machine learning to enable efficient and accurate PDE solvers.