The Radio Frequency Spectrometer (RFS) is a two-channel digital receiver and spectrometer, which will make remote sensing observations of radio waves and in situ measurements of electrostatic and electromagnetic fluctuations in the solar wind. A part of the FIELDS suite for Solar Probe Plus (SPP), the RFS is optimized for measurements in the inner heliosphere, where solar radio bursts are more intense and the plasma frequency is higher compared to previous measurements at distances of 1 AU or greater. The inputs to the RFS receiver are the four electric antennas mounted near the front of the SPP spacecraft and a single axis of the SPP search coil magnetometer (SCM). Each RFS channel selects a monopole or dipole antenna input, or the SCM input, via multiplexers. The primary data products from the RFS are autospectra and cross spectra from the selected inputs. The spectra are calculated using a polyphase filter bank, which enables the measurement of low amplitude signals of interest in the presence of high-amplitude narrowband noise generated by spacecraft systems. We discuss the science signals of interest driving the RFS measurement objectives, describe the RFS analog design and digital signal processing, and show examples of current performance. Plain Language Summary Solar Probe Plus (SPP) is a NASA mission which will travel much closer to the Sun than any previous spacecraft. The FIELDS experiment on SPP is composed of sensors (antennas and magnetometers) and receivers which will measure the electric and magnetic fields in this unexplored region. This paper describes the Radio Frequency Spectrometer (RFS), a receiver which will measure radio waves up to 19.2 MHz. We describe the types of radio signal sources present in interplanetary space close to the Sun, show how the design of the RFS makes it possible to measure these sources, and demonstrate the current performance of the receiver.