This chapter addresses the need to design non-linear methods that circumvent Godunov’s theorem for solving hyperbolic partial differential equations. After providing some background on classical polynomial interpolation theory, we examine modern, non-linear polynomial interpolation methods. These methods determine polynomial coefficients based on the behaviour of the data. Such polynomials are designed to handle not only smooth data but also rapidly changing data, even discontinuities. We begin with the standard ENO (Essentially Non-Oscillatory) method, along with two modifications that produce conservative ENO schemes. One modification is classical and based on the primitive function, while the other is more direct and conceptually simpler, relying on enforcing conservation on a preselected ENO stencil. Next, we introduce the classical WENO (Weighted Essentially Non-Oscillatory) methodology, along with a more recent variation. These non-linear spatial reconstruction methods are introduced for a single function. We then explore extensions of these methods to vector functions. The simplest extension is the component-by-component approach, in which the scalar reconstruction scheme is applied to each component of the vector of unknowns, typically the conserved variables. However, collective experience suggests that this simple approach is inadequate for problems involving discontinuities or large spatial gradients, particularly for schemes of accuracy higher than two. A more successful extension involves the use of characteristic variables. Although the primary application of these methods is in the design of non-linear, high-order numerical schemes that attempt to circumvent Godunov’s theorem to solve hyperbolic partial differential equations, in this chapter we evaluate their performance purely as interpolation techniques. The chapter concludes with a summary, a brief review of methods not covered here, and suggestions for further study. A list of illustrative exercises is also provided.