BBC: The Front National's victory in the EU elections - its greatest triumph ever - is attributed to the policy of "detoxification" conducted by Marine Le Pen since she took over in 2011.
But its enemies remain unconvinced that the FN has become in any sense a normal or acceptable political force.
Mainstream politicians regard the FN as a classic party of the nationalist extreme, exploiting economic distress to whip up hatred of the outsider, the immigrant.
For the German Finance Minister Wolfgang Schauble, reacting to Sunday's vote, there is no question but that the FN is "fascist and extremist".
And in the UK, Nigel Farage of UKIP says he will never go into alliance with Marine Le Pen because of her party's "nasty, anti-Semitic past". » | Hugh Schofield, BBC News, Paris | Thursday, May 29, 2014