We study a class of orbit recovery problems in which we observe independent copies of an unknown element of $\mathbb{R}^p$, each linearly acted upon by a random element of some group (such as $\mathbb{Z}/p$ or $\mathrm{SO}(3)$) and then corrupted by additive Gaussian noise. We prove matching upper and lower bounds on the number of samples required to approximately recover the group orbit of this unknown element with high probability. These bounds, based on quantitative techniques in invariant theory, give a precise correspondence between the statistical difficulty of the estimation problem and algebraic properties of the group. Furthermore, we give computer-assisted procedures to certify these properties that are computationally efficient in many cases of interest. The model is motivated by geometric problems in signal processing, computer vision, and structural biology, and applies to the reconstruction problem in cryo-electron microscopy (cryo-EM), a problem of significant practical interest. Our results allow us to verify (for a given problem size) that if cryo-EM images are corrupted by noise with variance $σ^2$, the number of images required to recover the molecule structure scales as $σ^6$. We match this bound with a novel (albeit computationally expensive) algorithm for ab initio reconstruction in cryo-EM, based on invariant features of degree at most 3. We further discuss how to recover multiple molecular structures from mixed (or heterogeneous) cryo-EM samples.