There has been much talk of alternative fuels to keep our classics going as petrol becomes ever less attractive. But after all is said and done, more has been said than done.
Here a a brief summary of my experience, it is quite possible I have made a mistake here or there so if you spot one do let me know.

LPG.
Mostly Propane with a splash of Butane, stored as a liquid under a pressure of about 5 bar in steel cylinders. The fuel is turned into a gas and the pressure regulator in an evaporator that is heated by engine coolant. The simplest way of feeding the gas into the engine is by using a gas carburettor, but these are quite restrictive and so peak power is significantly reduced, maybe by 20%, although most drivers don't notice this. Often these systems are set up to run very rich to mask installation problems. Better way is to use gas injectors, similar to petrol injectors, with their own ecu, this is more expensive but more efficient.
LPG is usually less than half the price of petrol per litre, and although the fuel consumption on gas is slightly worse there is still an overall saving. A well set up system will have lower HC and CO emissions than petrol, but the higher fuel consumption leads to similar CO2 levels.
LPG has a higher octane rating and so engines can be redesigned with higher compression ratio to achieve higher power and economy, but again that is expensive.
As LPG is a dry fuel there can be issues with valve seats wearing, but only if the engine is worked hard. Special lubrication systems are available but many have problems with uneven distribution between the cylinders.
LPG supply is directly related to oil supply, so if petrol runs out then so does LPG.

Methane / CNG.
This is the same gas as you get in your domestic cooker, Compressed Natural Gas. On cars it is stored in very high pressure cylinders. You can get domestic pumps to fill the car from a household supply. The regulator and engine conversion equipment s similar to LPG, but with different flow rates. Again there is a power loss, slightly greater than LPG.
Also the valve wear issues are similar to LPG.

Methanol.
An alcohol fuel with a fairly low energy content, so less power than petrol. Very hydroscopic so cant be stored for long periods.

Ethanol.
An alcohol fuel with a fairly high energy content and octane rating, it makes a small gain in power on standard engines but can make even better gains on a modified engine with higher compression ratio.
It is very hydroscopic so has a limited tank life, it also has an unfortunate tendency to react with steel creating an acid that corrodes pretty much everything. So stainless or plastic fuel pipes, tanks and fittings are essential. The exhaust is also corrosive so stainless systems are best.
Alcohol fuels can be brewed in much the same way as moonshine, using food waste, but getting the water out is very difficult.
Flow rate is much higher than petrol, so the fuelling system has to be modified to suit, although some conversions have managed to use standard injectors and fuel pumps.

Hydrogen.
It has no carbon and the only emission is water. Unfortunately its very difficult to store, requiring cryogenic tanks. As it is the smallest atom it tends to leak through pretty much anything, including steel. However, CNG conversion equipment has been successfully used to convert the engine to run on hydrogen.
Making hydrogen from water using electricity is the easiest option, but home made equipment is often very dangerous. Companies such as ITM Power offer home hydrogen generators. If powered from your own wind turbine it gives free fuel.
Again it's a dry fuel so valve seat erosion can be a problem if worked hard. Hydrogen takes up a lot of space in the intake so reducing air flow and power by a big margin, however the higher octane rating allows it to benefit fro turbocharing easily.

Biodiesel.
Another brewed fuel, usually made from oily seeds. Quite expensive to produce on anything other than large scale industrial quantities.
Performs just the same as regular diesel as long as it is filtered properly. Because of the greater solvent content it will loosen any old fuel deposits from an old cars fuel system, which can clog filters and injectors, so its best to fit a new fuel filter then run biodiesel briefly before changing the filter again.
Being an organic substance it is prone to stuff growing in it, so it has a limited tank life.

Chip oil.
Some people run older diesels on filtered chip oil, there are lots of microscopic plant particles in this which carbon up the injector tip and can coke up the engine. The harder the engine works the worse the problem gets. Using modern diesel can wash some of the deposits away, so alternating fuels can extend service life. Modern diesels use much finer tolerances in the injection system and are a lot less tolerant of this type of fuel.

Electric.
Throw the engine away and fit a milk float motor? It has been done but you have the performance of a milk float and the batteries are very heavy and take up a lot of space.
However, recently the century of pitiful investment has come to an end and the technology is finally getting proper funding. So over the next few years expect to see much more viable electric drive systems and batteries.
Personally I would love to develop a replacement electric system in the shape of an engine, I think that would be rather fun.

Advanced fuels from petrol stations.
Things like BP Ultimate and Shell V Power are formulated to coat the cylinder in a low friction coating, improve oil control and disperse more effectively to improve flame efficiency.
This does actually work, an engine improves over time with use as the coating builds.
The fuels cost more which can negate the economy benefit, depending on the engine and how much difference the fuel makes on that particular design.

Fuel supplements>

Hydrogen in petrol.
Mixing about 3% by volume hydrogen in the intake air can allow a petrol engine to run very lean, in this condition it is possible to gain 25% in efficiency. Maximum power will be reduced due to the displaced air, and the hydrogen requirement is quite large so generating it by an on-board electrolysis system is not usually viable. Also the engine change to make use of the lean burn potential is significantly expensive.
Introducing hydrogen without running the engine lean makes no improvement.
Using a jam jar of water and 12v electrodes is just plain silly.
A variation on this theme is to use thermal and catalytic cracking on the petrol to turn it into hydrogen and CO2, with engine modifications this can improve efficiency by up to 30% at part load. But its a very expensive system and complicated to control.

LPG in diesel.
Can improve efficiency by up to 20% but makes an even more dramatic improvement to performance. Over doing it will melt the engine, much like nitrous oxide on a petrol engine.
Cheaper than a conventional LPG conversion, this works very well indeed if set up properly.

The following things really don't work.
Magnets, they have no effect on fuel what so ever.
Solid fuel 'catalysts', I have tested a few and none made any difference.
Air flow vortex or turbulators, these usually have absolutely no effect, although they do block the intake and turbulent air reduces flow rates, so they can actually reduce performance. Cars with mass air flow meters require extremely non turbulent flow, and usually have flow straighteners built in, turbulent flow creates false readings and may make the engine run slightly rich.