A satellite system may have a constellation of communications satellites in orbits such as highly inclined eccentric geosynchronous orbits and low earth orbits. Satellite terminals may be used to communicate with the satellite constellation. The satellite terminals may have control circuitry that dynamically adjusts phased antenna array circuitry to steer antenna beams towards one or more satellites. Multiple antenna beams may be steered in different directions simultaneously. A satellite terminal may be used in simultaneously transmitting and receiving data from different respective satellites and may be used in transmitting and receiving satellite signals in multiple satellite bands. The satellite terminal may have an outdoor unit that is coupled to an indoor unit over a digital communications path. The outdoor unit may include the phased antenna array circuitry and transceiver and modem circuitry, whereas the indoor unit may cache media and serve as a firewall, router, and wireless access point.