With a rate of only 33.4 crimes per 1000 inhabitants, the Basque Country has one of the lowest crime rates in Europe. Even in big cities, only a few areas San Francisco in Bilbao, for example are best avoided. Although violent crime is extremely rare, the usual precautions still apply; avoid being alone at night in side streets, keep an eye on pickpockets, and you should do just fine.
The chances of being affected by ETA are extremely low. The number of ETA attacks has decreased a lot in the past few years 'only' 10 people have been killed by ETA in the past 4 years, 4 of them in the Basque Country, they are usually directed against one person, not many people or any building, and the targets are usually policemen, military, businessmen or politicians. ETA has never targeted visitors to the Basque Country.
Although they have also become rarer, there is a chance of seeing acts of vandalism related to radical pro-independence movements, specially if you visit the Basque Country during a big pro-independence-demonstration day or politically relevant dates the Gudari Eguna, the Aberri Eguna and some city festivals. Don't be alarmed, if you find yourself in the middle of a fight between the police and violent manifestants: do what the basques do, and go to the next street. In 30 min, everything will be probably finished.
It is advisable not to wear any Spanish symbols Spain's national football (soccer team T-shirts, Spanish flags or Real Madrid and Atletico Madrid club paraphernalia). Although no assaults on people wearing them have been recorded, some people may find them offensive or disrespectful.
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Basque politics are complex, very complex. Basques are divided between those who want full independence from Spain, those who ask for more self-government and those who think that the current union with Spain is just fine. And to this division you have to add the usual right VS left dimension.
Take into account that with a population of just 2 million people there are now 7 different political parties in the Basque Parliament and that one party has been illegal for the last 10 years.
So if speaking to strangers, avoid political topics as it can cause upset or offense.