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Satellite quantum communications are emerging within the panorama of quantum technologies as a more effective strategy to distribute completely-secure keys at very long distances, therefore playing an important role in the architecture of a large-scale quantum network. In this work, we apply and extend recent results in free-space quantum communications to determine the ultimate limits at which secret (and entanglement) bits can be distributed via satellites. Our study is comprehensive of the various practical scenarios, encompassing both downlink and uplink configurations, with satellites at different altitudes and zenith angles. It includes effects of diffraction, extinction, background noise and fading, due to pointing errors and atmospheric turbulence (appropriately developed for slant distances). Besides identifying upper bounds, we also discuss lower bounds, i.e., achievable rates for key generation and entanglement distribution. In particular, we study the composable finite-size secret key rates that are achievable by protocols of continuous variable quantum key distribution, for both downlink and uplink, showing the feasibility of this approach for all configurations. Finally, we present a study with a sun-synchronous satellite, showing that its key distribution rate is able to outperform a ground chain of ideal quantum repeaters.