What are the advantages/disadvantages of diesel/Jet A-1 piston engine?
Advantages

Compression-ignition engines have higher compression ratio which leads to being more efficient. So the aircraft range improves and on longer flights payload capacity may improve as well. Fuel consumption is usually about 30% lower.
The higher efficiency combined with the fact that Jet A-1 fuel is cheaper mean they have lower operating cost. Especially since in Europe the leaded avgas is heavily taxed to discourage it's use for environmental concerns.
They should be a bit more reliable. A diesel engine needs high pressure fuel pump, but apparently there are less problems with those than there are with spark-plugs.
They are somewhat easier to operate, since they don't have separate throttle and mixture controls.
They don't suffer the problems associated with incorrect settings of those controls, namely knocking/detonation and pre-ignition.
Turbo-charging them does not carry the risk of pre-ignition and most of them are turbo-charged, so the performance does not decline as fast with density altitude.
There is lower risk of fire since jet/diesel fuel is less flammable (has higher ignition temperature).
Jet A-1 has much wider availability, especially in the third world, so it's easier to plan for fuel stops on a flight through say, Latin America or Africa

Disadvantages

They are heavier for the same power, because they need to be made from stronger material due to the higher compression ratio and because they need to have larger cylinder volume because of lower maximal rpm. On longer flights the reduction in fuel weight often makes up for the heavier engine.
Turbo-charging comes with specific operating procedures unfamiliar to those used to normally aspirated engines and a slight lag in thrust lever (it's not a throttle) response.
As others mentioned, small airports may not have jet fuel yet. This is probably better in Europe where the pressure to phase out 100LL avgas is stronger. Some engines also use automobile diesel fuel or even either jet or diesel fuel as diesel engines are less picky about what they burn.