The large majority of the circuits used are resonant converters (a.k.a Royer converters; cf. Bright, Pittman and Royer, “Transistors As On-Off Switches in Saturable Core Circuits,” Electrical Manufacturing, December 1954.). A pulsed current through a transformer is back-fed to the driving transistors' base connections via auxiliary windings on the same transformer.
This answer to a a question about the special transformers used in these resonant converters provides plenty of links to good sources for further reading. Compact fluorescent lamps (CFLs) use a very simple yet elegant type of these circuits, where the saturation characteristics of the core determine the power output to the lamp, while most LCD backlight circuits of computer monitors or laptops use this circuit with a means of electronic pre-regulation, as designed by Jim Williams (1948-2011) and documented as US patents No. 5,408,162 and 6,127,785 and Linear Technology application notes AN49, AN55 and AN65. This concept was further developed into using piezoelectric transformers, cf. AN81.
There are also circuits using an oscillator running at a fixed frequency and a transformer to step-up the voltage to the lamp's requirements. Often, a 555 (timer IC) is being used as a rudimentary low-frequency oscillator, providing a pulse train to the transistors which are switching the primary of the transformer, giving you AC output from its secondary. An example of this sort of circuit is liked here.
Note: I have borrowed this information from Madmanguruman's answer to the now-closed repair question, not because I want to steal his fame/reputation, but because I believe the information is valuable and should be preserved in a non-closed question.
In addition, circuits exist that are in between the resonant and fixed-frequency oscillator concepts. By looking at the board of a commercially available emergency lamp, ...
... I tried to extract this schematic. Please note it is not complete and covers only the components between the oscillator IC (555 timer) and the transformer:
The output stage would look simpler if a complementary transistor pair would have been used (npn and pnp), or if one rectangular driving voltage would go to one npn power transistor and, inverted by another small transistor, to the second npn power transistor, but it seems the designers decided to stick with one type of transistor only or not use an extra phase-inverting transistor - at the cost of using an additional winding on the transformer. Here's what the circuit does:
The IC's open collector output drives transistor Q6 via a 2k4 resistor. I assume the voltage at Q6's collector is designed to be quite rectangular, i.e. the transitions from high to low and back to high should not be slow. While the transistor inside the IC is still off, Q6 is off because its base is pulled high. Once the transistor in the IC turns on, Q6 turns on as well and feeds base current into Q8. This causes two things to happen: Current flows through the 1st winding of the transformer (S1 becomes low with respect to F1), and Q7 is kept in the off state because just as S1 is lower than F1, S3 is lower than F3. Therefore, at the same time that Q8's base goes high, Q7's base goes low.
If, after all this, the IC's output goes high again, Q6 turns off, and the collector current through Q8 will shut off, too. The energy stored in the transformer wants to go somewhere, however, and this will cause all (!) windings to reverse their polarity: S1 starts high with regard to F1, S3 will also start high with regard to F3, Q7 turns on because its base is driven high by S3-F3, F2 will dive below S2, and of course, the output winding (S4-F4) will also reverse its voltage, thereby creating an AC output for the lamp.
This state seems to be held up by the energy stored in the transformer and in the inductor above and the capacitors beneath the primary windings.
From there, the process starts over again as soos as the timer IC initiates the next cycle of the AC output signal; it appears that the frequency at the IC's output should be designed to match what the transformer and the components around it are designed to do.
It looks like the circuit is operating somewhere in between a purely puls-width-driven mode, where the timer IC would be the only part that says when the power transistors Q7 and Q8 are on or off, and a purely resonant mode, where the transformer and the capacitors around it have the authority to drive Q7 and Q8, because then, we would need yet another winding driving Q8's base. My understanding is that the 555 initiates each cycle and the resonant components (L, C, transformer) determine when the cycle stops in case the IC is not faster anyway. Using LT Spice, I found that this circuit might work at a frequency of maybe 500 Hz...3 kHz.
Note: While weird in the sense of traditional Q&A sites, SE encourages creating and answering your own questions for the sake of getting useful information onto the site, in the sense of a wiki.