Google and other search engines are obligated to honor valid requests to remove content under a search for a person's name due to a set of European laws known as the "Right To Be Forgotten", but a part of the interpretation of those rules is being called into question in the face of French authorities moving to uphold Google's recent decision to deny four such requests. Specifically, courts are trying to decide if information that could be detrimental to a person despite being true, such as political leanings and criminal records, should be accessible to the general public via a simple web search. It's essentially a case of privacy and possible detriment versus public interest. The case has yet to have a date set for a hearing as of this writing.
The case that kicked it all off was an appeal sent to CNIL, the French privacy administration, by four disgruntled individuals who had made Right To Be Forgotten requests of Google. The requests were deemed invalid by Google, the reasons for which can include that the information is accurate and not unnecessarily painting the individual in a poor light, among other things. This denial led to an appeal, which was initially shot down, but was sent to the European Court of Justice out of Luxembourg. What the case boils down to at this point is whether links containing sensitive personal information, as outlined above, should be delisted from the search engines on request, or if it's in the overall public interest to leave them up.
According to the ECJ, at the very least, search engines being compelled to remove such content without a relevant review for public interest on a per-case basis risks the creation of a condition wherein individuals can simply claim that a link contains sensitive information and have it removed, regardless of its value to the public. The case in point has to do with a video that shows personal relationships within the Church of Scientology among high-ranking staff, including public relations staff. The links in question also consist of articles alleging that Church of Scientology members committed or had links to specific serious crimes.