SUMMARYWe recall and discuss the theory of a 2-hop downlink in deep-space communications. The first hop (length d 1 ) links the deep-space spacecraft to either a geostationary satellite or to a low Earth orbiting satellite. The second hop (length d 2 ; with d 2 {d 1 ) links the satellite to the Earth receiver through a transparent or regenerative transponder, in slant paths affected by troposphere attenuation. If we adopt a BPSK or QPSK modulation scheme and the same carrier frequencies in the two hops with a transparent transponder, a particular value of the carrier frequency makes the noise-to-signal ratio minimum. A better choice is to assign a low carrier frequency (X-band) to the second hop and a high one (W-band) to the first hop. A 2-hop downlink is superior to a 1-hop downlink with a large power gain proportional to ðd 1 =d 2 Þ 2 c1 at high microwave frequencies and large troposphere attenuation (high link-availability). Shannon's capacity theorem provides the same large gain independently of the choice of the carrier frequencies in the two hops, if we use a regenerative transponder. The carrier frequencies of a 2-hop downlink need not be those reserved to deep-space communication, because the spacecraft could communicate as if it were transmitting 'from the Earth' through a conventional satellite connection. We have applied the theory to a first-order design in the frequency range 10-100 GHz, slant path elevation angles 308; 458 and 908 at Gera Lario or Fucino (Italy), and downlinks form Mars and Saturn, although the general findings and methodology are of global applicability.