Spin-orbit torques (SOT) in thin film heterostructures originate from strong spin-orbit interactions (SOI) that, in the bulk, generate a spin current due either to extrinsic spin-dependent, skew, or/and side-jump scattering or to intrinsic Berry curvature in the conduction bands. While most SOT studies have focused on materials with heavy metal components, the oxide perovskite SrRuO3 has been predicted to have a pronounced Berry curvature. Through quantification of its spin current by the SOT exerted on an adjacent Co ferromagnetic layer, we determine that SrRuO3 has a strongly temperature (T)-dependent spin Hall conductivity σSH, increasing with the electrical conductivity, consistent with expected behavior of the intrinsic effect in the "dirty metal" regime. σSH is very high at low T, e.g., σSH > (h/2e)3 × 105 ω-1 m-1 at 60 K, and is largely unaffected by the SrRuO3 ferromagnetic transition at Tc ≈ 150 K, which agrees with a recent theoretical determination that the intrinsic spin Hall effect is magnetization independent. Below Tc smaller nonstandard SOT components also develop associated with the magnetism of the oxide. Our results are consistent with the degree of RuO6 octahedral tilt being correlated with the strength of the SOI in this complex oxide, as predicted by recent theoretical work on strontium iridate. These results establish SrRuO3 as a very promising candidate material for implementing strong spintronics functionalities in oxide electronics. © 2019 American Chemical Society.