How Military to Civil Aircraft Transition is made by National Authorities?
In the US, ex-military aircraft must be issued a special airworthiness certificate before they can be flown as civilian aircraft. Some aircraft may be denied the certificate outright, but most can be flown this way. The hard part is maintaining them, the cost of operating them, and finding qualified people to crew them, including a qualified flight instructor and even a Designated Examiner if the FAA requires a type rating to fly it.

In some very rare cases, it is possible to get a standard or utility airworthiness certificate, but the aircraft must have been previously approved that way before you can do that. 

In the majority of cases, the aircraft is given a special airworthiness certificate under the Experimental category, Exhibition subcategory. This means the aircraft can only be operated to exhibit the aircraft’s flight capabilities, performance, or unusual characteristics for air shows, motion pictures, television, and similar productions, and for the maintenance of exhibition flight proficiency. To fly it, you must meet certain experience requirements and in some cases you need to obtain a pilot type rating for that aircraft. 

In other cases, if you want to race the aircraft in air races, you can get a special airworthiness certificate for that purpose, to operate the aircraft in air races, practice for air races, and to fly to and from racing events.

To get the airworthiness certificate in the first place you have to submit a maintenance plan for the aircraft to the FAA (often derived from maintenance plans others have drafted) and have it approved by the FAA, and the aircraft must be subject to regular inspections as per the plan. 

Source: FAA Experimental Airworthiness Certificates home page

In yet other cases, some military cargo aircraft are converted for missions such as firefighting. Those are exceptional cases, almost unique to the US in that they are not required to comply with many federal aviation regulations.


  The Federal Aviation Administration (FAA) has abrogated any
  responsibility to ensure the continued airworthiness of "public-use"
  aircraft, including ex-military aircraft converted to firefighting air
  tankers. Although these aircraft are awarded FAA type certificates,
  the associated certification processes do not require testing and
  inspection to ensure that the aircraft are airworthy to perform their
  intended missions. Source


For other countries I suggest you list the ones you are interested in. I only know the rules for the US/FAA.