Apparatus for the wireless transmission of data, and preferably also of power, across a space between a length of production tubing and a surrounding casing in a petrochemical well, includes a pair of inductively-coupled coils, a first of which is located on the exterior of the production tubing generally coaxially therewith, and the second of which is located on the interior of the casing generally coaxially therewith. This may be used in particular as part of a system for transmitting power and data to/from a sensor monitoring the pressure and/or other environmental conditions within the “B” annulus B of a sub-sea well.