Abstract: The Lyman-alpha (Lya) forest is currently one of the most powerful probes of large-scale structure (LSS) at high redshift (z > 2). This is made possible by large spectroscopic surveys that measure hundreds of thousands of quasar spectra, such as the Baryon Oscillation Spectroscopic Survey (BOSS) and its successor, extended BOSS (eBOSS). However, over the last decade 3D correlations of the Lya forest have only been used to measure baryon acoustic oscillations (BAO). In this talk, I will present the first ever cosmological analysis of the full shape of the Lya 3D auto-correlation function, and its cross-correlation with quasars. We use Lya correlations measured by eBOSS using data release 16 (DR16) and fit the full shapes of these correlations in order to measure the Alcock-Paczynski (AP) effect. Our measurement delivers a factor of two improvement in constraining power compared to BAO analyses. Finally, I will show how constraints on the Hubble constant and dark energy are improved by using the full shapes of the Lya correlations.