Eurosceptic trend in Denmark and Greece
15 May 2014 Last updated at 03:11 BST
In troubled Greece the opinion polls reflect voter discontent, with the radical left-wing Syriza bloc now neck-and-neck with the ruling conservative New Democracy.
Both are on just over 26% each, so Syriza is likely to enter the European Parliament for the first time - and in strength.
In affluent Denmark the People's Party (DF), with a nationalist and Eurosceptic outlook, is topping the latest polls. Centrist parties are second and third.
Chris Morris reports from Athens and Matthew Price from Copenhagen.