What is the chemistry behind this "petrol from air" technology? A recent news report in the UK claimed a breakthrough in making a petrol equivalent from carbon dioxide and water:


  A small British company has produced the first "petrol from air" using a revolutionary technology that promises to solve the energy crisis as well as helping to curb global warming by removing carbon dioxide from the atmosphere.


It sounds like the news reports are over-hyping it a little (and I'm not sure it is that new), so what is the real chemistry behind it? What are the key reactions and how much input energy is required?
Step one: trap $\ce{CO2}$. This can be done using specially selected amines, like triethanolamine. Aminosilicones (a common chemical used in hair conditioner), can also do this.

Step two: release trapped $\ce{CO2}$. In case of triethanolamine you'll have to boil mixture.

Step three: mix $\ce{CO2}$ with hydrogen (hydrogen is produced by electrolysis of water). Volume proportion should be somewhere 1:3. 

Step four: threat the mixture with proper catalyst. This step may be divided into two stages: reverse water-shift reaction  and some variant of Fischer–Tropsch process with some type of alcohol synthesis as very interesting candidate. There are some multi-functional catalysts that can perform both stages at once.

Another variation on step 4 is the Sabatier process, which involves generating methane from $\ce{CO2}$ and hydrogen.  Typically the catalyst is Rainy nickel (powder).

The required energy in form of electrical current is probably from 1.5 to 3 times of heat of combustion of resulting fuel, which is in turn at least twice more that can be converted into actual work again.