Sprites, blue jets, and elves
The preferred term for these types of events is Transient Luminous Event (TLE). 

In a 1924 paper, C.T. Wilson suggested that "The electric field of the [thunderstorm] cloud may cause ionization at great heights..."  "At a height of 60 km, the density of the air is about 1.6 x 10^-4 of that near the ground, while the composition of the air is not very different, so that the critical value of the field may be taken as about 30,000 x 1.6 x 10^-4 = 4.8 volts per centimetre."

Despite many reports of optical phenomena high above thunderstorms (often from pilots flying at night), credit for the first photographic evidence of these kinds of phenomena is given to John Winckler, a researcher at the University of Minnesota.  On the night of July 6, 1989 he was testing a low light level video camera and later found two images of sprites on the video he had recorded (ref).

Here's a pretty good figure summarizing the various types of luminous event (source: http://smsc.cnes.fr/TARANIS/GP_science.htm; CNES is the French Centre National d'Etudes Spatiales)



Sprites
The following figure is adapted from a review article by C.J. Rodger (1999) and shows some of the forms sprites may take:

Color	red
Minimum altitude	50 km (tendrils < 40 km)
Maximum altitude	90 km
Width (km)	25 - 50 km (columns >2 km)
Duration	5 - 300 ms
Association with lightning	positive cloud-to-ground (ELF slow tail)
Occurrence rate	1 in 40 to 1 in 2.3

Sprites are dim and hard to see with the naked eye (adapted from http://elf.gi.alaska.edu/#intro)

To see a sprite you need an unobstructed view of the region above an active thunderstorm.  The best viewing distance is about 100 - 200 miles from a storm.
The dark adapted eye most readily sees sprites in parfoveal vision (looking out the corner of your eye (rods) rather than looking directly at the sprite (cones which respond better to colors).

It must be very dark, no city light, no twilight.  Cloud illumination from lightning activity may be too bright or may be distracting.

Sprites are very brief (3-10 ms usually).  They are produced by only about 1% of lightning strokes.

Sprite videos:
sprite movie (Univ. AK, Geophys. Inst.)
sprite in slow motion (H.H.C. Stenbaek-Nielsen, U. Alaska Fairbanks, DARPA, NSF)
sprite movie high speed video (NM Tech)
sprite movie high speed video (NM Tech)
here are some spectacular pictures and videos (not sure how to cite this page)

Some of my favorite pictures are still images (especially ones that include the foreground and surroundings)
Mike Hollingshead (Astronomy Picture of the Day)
Sprite with Aurora (Walter Lyons and an Astronomy Picture of the Day)
A hard to spot sprite viewed from the International Space Station (ISS Expedition 31 crew, NASA)
sprite (National Geographic) (click here if the link is slow to respond)
sprite (Geospace Physics Laboratory, Florida Inst. of Tech.)
sprite (WDRB.com)
sprite (Weatherscapes)

Blue jets
These begin and extend upward from the tops of thunderstorms.  Characteristics of blue jets and blue starters are shown below (from Rodger 1999).

It is still not clear what produces blue jets, they do not necessarily follow directly after a lightning discharge.  The propagate upward at about 1 x 10⁵ m/s.  The deep blue color, which is more readily scattered by air, makes them difficult to see and photograph.  Some characteristics of blue jets are shown below (from Rodger, 1999)

Color	deep blue
Minimum altitude	about 20 km
Maximum altitude	40 to 50 km
Width (km)	about 3 km (cone about 15 degrees)
Duration	about 250 ms
Occurrence rate	2.8 per minute (in 22 min.)

blue jet videos
blue jet movie (Univ. AK, Geophys. Inst.)

still photographs
blue jet (PBS Nova)
Stanford University photograph (this might be an example of a giant blue jet)
blue jet photographed above a thunderstorm in the Northern Territory, Australia (Thijs Bors published in The Telegraph)
blue jet on St. Barth (credit: Elka Liot, Muskapix Gregory Moulard.  "St. Barth" refers to Saint Barthelemy a French Overseas Collectivite Territoriale in the Carribean.  Together with Saint Martin, Guadeloupe, and Martinique it is part of the French West Indies)

Elves
The name is an acronym for Emission of Light and Very Low Frequency Perturbations Due to Electromagnetic Pulse Sources.

Color	. . .
Minimum altitude	75 km
Maximum altitude	105 km
Width (km)	100 - 300 km
Duration	< 1 ms
Association with lightning	intense positive cloud-to-ground (average 148 kA)
Occurrence rate	. . .

Videos of Elves
Blue jets, Sprites, and Elves (New Scientist video)

Production of X-rays and γ-rays by lightning
This section needs some updating

Moore et al. (2001) have reported observing bursts of high energy radiation associated produced in a 1 to 2 ms interval just before the start of the return stroke and as the stepped leader was nearing the ground.  Dwyer et al. (2003) observed bursts of energetic radiation in the last 160 μs of the dart leader and possibly right at the start of the return stroke in 31 out of 37 triggered events studied.  The shorter interval may be because dart leaders have a higher propagation speed.  My understanding is that the sensors used in both these two experiments were unable to distinguish between energetic electrons, X-rays, and gamma rays.

A NaI (sodium iodide) scintillation detector was used in both experiments.  Light is emitted when the high-energy radiation strikes the NaI.  The weak light signal is then detected and converted into an electrical current using a photomultiplier tube (source of the figure below)



A sketch of the signal produced by the scintillation detector used by Dwyer et al.,  when exposed to a Cesium-137 γ-ray source, is shown below (the control sensor was identical except it did not include the NaI scintillator). 


The exponential decay following the peak is the response of the preamplifier circuit. 

The figure below is a sketch of high energy radiation produced by a triggered lightning discharge from the Dwyer et al. (2003) paper.


The output of the scintillation detector is shown at top.  This signal began 160 μs before the start of the return stroke (the return stroke began at time 0 in the figure).  The complex signal shape during the rise to peak indicates that several energetic particles were detected.  A very small pulse is just visible on the control signal trace and may have been caused by energetic particles directly striking the photocathode of the photomuliplier tube.  The return stroke current was measured at the strike point and rose to a peak value of about 22.5 kA.  The electric field was measured at a point 260 m away from the strike point and has the characteristic asymmetric V-shape described by Rubenstein et al. (1995) for the leader-return stroke transition observed at close range.

An improved sensor was used during the summer 2003 campaign and Dwyer et al. (2004) report X-rays were measured 0 to 80 μs prior to and at the beginning of 73% of triggered return strokes studied.  Each X-ray burst usually lasted less than 1 μs.  The most intense bursts come from parts of the channel that is within 50 m of the ground.  A sketch of one of the recorded signals is shown below.  The return stroke began at time t=0.


The lightning struck about 50 m from the electric field derivative antenna (the E field signal above is an integration of the measured dE/dt signal) and about 260 m from the x-ray detector. 

It seems clear that the x-rays are produced during the stepping process.   Because of the similarity between x-rays produced by dart leaders in triggered lightning and stepped leaders in natural lightning Dwyer et al. (2005) suggests that the production mechanisms are similar and that dart leaders also step, but with a frequency that isn't resolved on optical or field records.

Implications
Emissions from dart leaders in triggered lightning and stepped leaders in natural lightning are similar.  This suggests some similarities in the discharge process (dart leaders may actually step)

Observations may provide some clues about leader propagation processes.

The standard "relativistic runaway electron avalanche model" might have some trouble trying to explain the lightning generated X-ray emissions.


Schumann resonance
Lightning signals excite the earth - ionosphere cavity.  Resonant frequencies are amplified. 

source of the image above	source of the plot above