Radio detection of cosmic ray air showers in the digital era

In 1965 it was discovered that cosmic ray air showers emit impulsive radio signals at frequencies below 100 MHz. After a period of intense research in the 1960s and 1970s, however, interest in the detection technique faded almost completely. With the availability of powerful digital signal processing techniques, new attempts at measuring cosmic ray air showers via their radio emission were started at the beginning of the new millennium. Starting with modest, small-scale digital prototype setups, the field has evolved, matured and grown very significantly in the past decade. Today's second-generation digital radio detection experiments consist of up to hundreds of radio antennas or cover areas of up to 17 km$^{2}$. We understand the physics of the radio emission in extensive air showers in detail and have developed analysis strategies to accurately derive from radio signals parameters which are related to the astrophysics of the primary cosmic ray particles, in particular their energy, arrival direction and estimators for their mass. In parallel to these successes, limitations inherent in the physics of the radio signals have also become increasingly clear. In this article, we review the progress of the past decade and the current state of the field, discuss the current paradigm of the radio emission physics and present the experimental evidence supporting it. Finally, we discuss the potential for future applications of the radio detection technique to advance the field of cosmic ray physics.