A real-time broadband radio frequency (RF) spectrum analyzer is proposed and experimentally demonstrated to rapidly measure the RF spectrum of broadband optical signal. Cross phase modulation in the highly-nonlinear fiber is used to convert the RF spectrum carried by the pump to the optical spectrum of the probe signal, then the optical spectrum is real-time analyzed with the parametric spectro-temporal analyzer (PASTA) technology. The system performances are investigated in detail, including bandwidth, resolution, frame rate, and dynamic range. It achieves large RF bandwidth of over 800 GHz, as well as 91-MHz frame rate without sacrificing the resolution. It is noted that 91-MHz frame rate is several orders of magnitude improvement over those previous reported all-optical RF spectrum analyzers. As a proof-of-concept demonstration, this real-time broadband RF spectrum analyzer successfully characterizes the ultra-short pulse trains with repetition rate of 160GHz, which is far beyond capability of the conventional electrical spectrum analyzer. It presents a new way to implement rapid and broadband RF spectrum measurement, and would be of great interests for some ultrafast scenarios, where the real-time RF spectrum analysis can be applied.