Gaussian processes (GPs) are popular models for functions, time series, and spatial fields, but direct application of GPs is computationally infeasible for large datasets. We propose a multi-scale Vecchia (MSV) approximation for modeling and analysis of multi-scale phenomena, which are ubiquitous in geophysical and other applications. In the MSV approach, increasingly large sets of variables capture increasingly small scales of spatial variation, to obtain an accurate approximation of the spatial dependence from very large to very fine scales. For a given set of observations, the MSV approach decomposes the signal into different scales, which can be visualized to obtain insights into the underlying processes. We explore properties of the MSV approximation and propose an algorithm for automatic choice of the tuning parameters. We conduct numerical studies and comparisons to existing approaches.