Stimulated Raman scattering spectroscopy is a powerful nonlinear optical technique for label-free identification of molecules, based on their characteristic vibrational fingerprint. Current implementations of SRS, while achieving extremely high acquisition speeds up to the video rate, mostly work at a single frequency, thus providing limited chemical information. Broadband implementation of SRS is technically challenging, as for imaging dilute species in biological microscopy applications one must detect very small (approx. 10^-5) signals sitting on a large background via modulation transfer technique. We introduce and experimentally demonstrate a novel approach to broadband SRS spectroscopy based on photonic time stretch (PTS). The broadband femtosecond Stokes pulse, after interacting with the sample, is stretched by a telecom fiber to 15ns duration, mapping its spectrum in time. The signal is sampled through a fast analog-to-digital converter, providing single-shot spectra at 80-kHz rate. We demonstrate 10^-5 sensitivity, over 500 cm-1 bandwidth in the C-H region with high resolution. These performances are already suitable for a number of applications, such as monitoring microfluidic flows, the onset of chemical reactions or solid-state samples such as pharmaceutical products. As the acquisition speed of PTS does not depend upon the covered spectral region, we are planning to extend the spectral coverage of SRS to the fingerprint region. Furthermore, using commercially available lasers with higher rep-rates, we could shorten the acquisition time considerably. This will pave the way to high-speed broadband vibrational imaging for materials science and biophotonics.