Using high resolution spectra from the VLT LEGA-C program, we reconstruct the star formation histories (SFHs) of 607 galaxies at redshifts z = 0.6 − 1.0 and stellar masses 10^10 M⊙ using a custom full spectrum fitting algorithm that incorporates the emcee and FSPS packages. We show that the mass-weighted age of a galaxy correlates strongly with stellar velocity dispersion (σ∗) and ongoing star-formation (SF) activity, with the stellar content in higher-σ∗ galaxies having formed earlier and faster. The SFHs of quiescent galaxies are generally consistent with passive evolution since their main SF epoch, but a minority show clear evidence of a rejuvenation event in their recent past. The mean age of stars in galaxies that are star-forming is generally significantly younger, with SF peaking after z < 1.5 for almost all star-forming galaxies in the sample: many of these still have either constant or rising SFRs on timescales > 100 Myrs. This indicates that z > 2 progenitors of z ∼ 1 star-forming galaxies are generally far less massive. Finally, despite considerable variance in the individual SFHs, we show that the current SF activity of massive galaxies (> L∗ ) at z ∼ 1 correlates with SF levels at least 3 Gyrs prior: SFHs retain ‘memory’ on a large fraction of the Hubble time. Our results illustrate a novel approach to resolve the formation phase of galaxies, and in identifying their individual evolutionary paths, connects progenitors and descendants across cosmic time. This is uniquely enabled by the high-quality continuum spectroscopy provided by the LEGA-C survey.