While not originally part of Japanese (the use of か helps distinguish between statements and questions), it is generally acceptable to use the question mark. This goes especially true with comics, novels and everyday conversation, where か is often omitted (for example: 「大丈夫？」).
The Ministry of Education has published guidelines of its usage in 1941.
To summarize the guideline:
1) Normally, you do not use it. However, you can use when appropriate in order to indicate the doubting tone of a spoken line.
example: ｢えゝ？ なんですつて？｣
2) Can be used in questions and cross-questions.
3) Can be used in comics to express wordless doubt.
Note though, that official documents and business correspondence do not use the question mark, but instead adhere to the traditional grammar rules.
user224579 has already posted the official name 疑問符, but it can also be referred to as クエッションマーク or はてなマーク.
edited to add a rough summary of the pdf.