Emergency Operations
Authority and squawk codes
14 CFR 91.3 gives the pilot in command authority to deviate from any rule in an emergency to the extent necessary, and to report the deviation on request. Squawk 7700 for an emergency, 7600 for radio failure, 7500 for hijacking. If radios fail entirely, light gun signals still apply: steady green cleared to land, flashing green return for landing, steady red give way, flashing red airport unsafe, alternating red and green exercise extreme caution.
Lost communications (91.185)
Route: last assigned, vectored, expected, or filed, in that order. Altitude: the highest of the last assigned altitude, the minimum IFR altitude, and the altitude you were told to expect. Then leave the clearance limit: if it is a fix from which an approach begins, begin the approach at the EFC or ETA; if not, proceed to such a fix at the EFC or ETA.
Other emergencies
For an emergency descent, turn off the airway, descend to the lowest safe altitude, squawk 7700, and advise ATC. Declare minimum fuel when you can accept no delay; declare a fuel emergency when the remaining fuel puts landing at risk. ELTs transmit on 406 MHz, with legacy units also on 121.5.
Traps
The lost-communications altitude is the highest of three values, not the last assigned, and diversion never relieves you of minimum IFR altitudes.
Worked example: last assigned 3,000 ft, MEA 5,000 ft, no EFC received, and comms fail. You climb and maintain 5,000 ft.