Kernel-based interpolation is a classical and versatile tool for reconstructing unknown functions from scattered measurements. The interpolant is computed by solving a linear system of size equal to the number of measurements. When using reproducing kernels associated with Sobolev spaces, one can leverage rigorous error and convergence theory. However, if the kernel additionally has compact support, a fundamental \emph{trade-off principle} arises: either convergence is achieved at the cost of ill-conditioned kernel matrices, or the matrices remain well-conditioned but convergence is lost. Multiscale methods circumvent this issue, producing schemes that are both convergent \emph{and} computationally efficient. The approach constructs a sequence of site subsets (via thinning algorithms or explicit construction) and associated scaling parameters, defining local approximation spaces as linear spans of kernel translates on each subset. The method then proceeds as a residual-correction scheme, iteratively refining the approximation. Unlike standard wavelet multiscale constructions, each local approximation space carries a distinct, but norm-equivalent, inner product, which prevents direct application of classical multiscale arguments and requires different analytical tools. Importantly, the resulting multiscale linear system has size growing only linearly with the number of measurements. Recent developments have enabled a precise characterization of the \emph{global approximation space}, defined as the sum of the local spaces, through the construction of a \emph{Lagrange basis} of this space. This basis allows a \emph{nodal representation} of the multiscale interpolant, which is valuable for applications, and offers a deeper understanding of the structure of the global approximation space. In this talk, I will introduce the kernel-based multilevel method, review established results, and present new findings on the Lagrange bases and their consequences for the kernel-based multilevel method.