Real-time spectroscopy provides invaluable information about the evolution of dynamical processes, especially non-repetitive phenomena. Unfortunately, the continuous acquisition of rapidly varying spectra represents an extremely difficult challenge. One method, wavelength-time mapping, chirps the spectrum so that it can be measured using a single-shot oscilloscope. Here, we demonstrate a method that overcomes a fundamental problem that has previously plagued wavelength-time spectroscopy: fine spectral resolution requires large dispersion, which is accompanied by extreme optical loss. The present technique uses an optically amplified wavelength-time transformation to beat the dispersion-loss trade-off and facilitate high-resolution, broadband, real-time applications. We show that this distributed amplification process can even be pumped by broadband noise, generating a wide gain bandwidth using a single pump source. We apply these techniques to demonstrate real-time stimulated Raman spectroscopy. Amplified wavelength-time Raman spectroscopy creates new opportunities for the study of chemical and physical dynamics in real time.