Understanding the internal structure of our planet is a fundamental goal of the earth sciences. As direct observations are restricted to surface outcrops and borehole cores, we rely on geophysical data to study the earth's interior. In particular, seismic reflection data showing acoustic images of the subsurface provide us with critical insights into sedimentary, tectonic, and magmatic systems. However, interpretations of these large 2D grids or 3D seismic volumes are time-consuming, even for a well-trained person or team. Here, we demonstrate how to automate and accelerate the analysis of these increasingly large seismic data sets with machine learning. We are able to perform typical seismic interpretation tasks such as mapping tectonic faults, salt bodies, and sedimentary horizons at high accuracy using deep convolutional neural networks. We share our workflows and scripts, encouraging users to apply our methods to similar problems. Our methodology is generic and flexible, allowing an easy adaptation without major changes. Once trained, these models can analyze large volumes of data within seconds, opening a new pathway to study the processes shaping the internal structure of our planet.