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Regrettably, there are many countries with legal or customary restrictions on the Bible. While a total ban is comparatively rare (e.g., North Korea punishes any possession of religious literature by death or imprisonment), it is more common for ownership or distribution to be limited:
Restrictions on attempts to convert members of other religions may also result in de facto bans on Bibles. Similarly, government actions in the name of "public order" may have the same effect, even if there is no law specifically banning Bible publication or ownership. In several cases, effective bans on religious freedom take place despite supposed constitutional guarantees to the contrary. Experiences may vary in different parts of the same country, or at different times.
So "illegal" is perhaps the wrong word, and we should instead be thinking of "Can anyone easily obtain a Bible in this country without attracting official difficulties?". Any such list will have fuzzy edges, of course. One indicator might be the ease with which Bible societies are able to do their work. They are organized groups with a missionary element and so the bar is higher for them than for private individuals.
The Gideons have a list of countries where they are not allowed to operate:
Afghanistan, Algeria, China (People's Republic), Comoros, Djibouti, Iran, Iraq, Libya, Maldives, Mauritania, Morocco, North Korea, Saudi Arabia, Somalia, Tunisia, Turkmenistan, Uzbekistan, Yemen
However, for some of these countries, other groups exist: there are United Bible Societies chapters for Algeria, Iraq, Morocco, and China, and some presence in Comoros, Djibouti, Mauritania and Somalia. The Catholic Bible Federation also operates in Iran. Not all of these efforts may be entirely government-approved.
For more detailed information, the US State Department issues annual reports on religious freedom with detailed assessments of the conditions on the ground in each country. In particular, they distinguish between what the country's law says, how the government acts, and how the wider society responds.