"We come to these elections calling for participation from all Spaniards in accordance to their convictions and principles," Abascal said. "We want to tell them that it is important that each Spaniard has their voice in the municipal and regional institutions, also in the European institutions as, often, this is not seen with much interest by citizens."
Vox erupted into the Spanish political scene in December when it won its first seats in the regional legislature for the Andalusia region. It then won 10% of the vote in national elections on April 28 to enter the Spanish Parliament.
Founded in 2013, Vox failed to win a seat in 2015 European elections, its first electoral test. Unlike other European far-right parties, Vox is not anti-European Union despite its ultra nationalist ideology.
Anti-EU and pro-EU parties are both hoping to make gains in a Brexit-dominated election for British seats in the European Parliament, while the governing Conservative Party is bracing for one of its worst-ever performances.
Daniel Hannan, a Conservative Member of the European Parliament, said he feared the party was facing "total wipeout" and would fail to gain any of Britain's 73 seats.
The Conservatives look likely to be punished for failing to take the country out of the EU as promised. Opinion polls suggest the newly founded Brexit Party led by Nigel Farage is set to take the biggest share of votes. There could also be a surge for the centrist Liberal Democrats, who want to stop Brexit.
In the last EU election in 2014, Farage's former UKIP party won 27% of the vote, helping to build momentum in the push to get Britain out of the EU.
Spaniards in the restive northeastern region of Catalonia are voting in European elections that include two high-profile separatist leaders who are running from jail and self-imposed exile.
Former Catalan regional president Carles Puigdemont and his ex-No. 2 Oriol Junqueras are both running on competing tickets of separatist parties for the European Parliament.
Junqueras is in jail in Madrid while on trial on charges that include rebellion for his part in Catalonia's attempt to secede from the rest of Spain in 2017. Puigdemont is wanted in Spain and fled to Belgium.
Both have been allowed to run as candidates, but would face legal hurdles to actually become European Parliament members if elected.
Polls and recent election results show that the 7.5 million residents of the wealthy Catalonia region are roughly split by the secession issue. It is highly unpopular in the rest of Spain.
"I think a lot is at stake for Catalonia in the European elections," Manuel Guajardo, a 57-year-old businessman, said after voting in Barcelona. "It will be the definition of a people, to take a path or another path, and these elections will mark this."
Barcelona is also holding municipal elections, where a separatist candidate is trying to oust its incumbent far-left mayor, who says she does not support secession but believes that Catalans should vote on the question.
In Castelbuono, a Medieval mountain town in Sicily's province of Palermo, a steady but sluggish stream of voters is showing up to cast ballots.
Many of the town's residents are backing the 5-Star Movement, a populist party now in a coalition government with the anti-migrant League party.
One of these is Vincenzo Messineo, a 32-year-old laborer who's concerned about youth unemployment and the influx of migrants.
"We don't want them all here," he said. "Europe is united so why can't they be divided among other countries too?"
For Anna Maria Ippolito, a 62-year-old 5-Star Movement supporter, financial inequality is a top concern.
"Six percent of the Italian population has all the wealth," she said. "It's not at all right. Paying the taxes that we do now just lets the rich get richer and the poor poorer."
On Europe, she thinks it's time to rein in the big guns. "Up to now, it's all been about Germany and France," she said. "They're the ones dragging us into this European disaster."
Migrants are on the mind of Silvia Bonomo, too, but she feels Europe needs to "open itself up" and do more to welcome people fleeing war and hardships. The 62-year-old middle-school teacher voted for the center-left Democratic Party.
"They wouldn't be coming if they didn't have a reason to," she said. "Migrants are seen as stealing jobs, which they are not. They're seen as criminals, which they are not. They are just like us."
People's Party Our Slovakia, a far right party that has 14 seats in Slovakia's parliament is expected to win seats in the European legislature for the first time.
The party openly admires the Nazi puppet state that the country was during World War II. Party members use Nazi salutes, blame Roma for crime, consider NATO a terror group and want the country out of the alliance and of the European Union.
The party received a boost in April after Slovakia's Supreme Court dismissed a request by the country's prosecutor general to ban it as an extremist group whose activities violate the Constitution.
Turnout in Slovakia at the previous vote in 2014 was 13%, the lowest in all EU countries.
The polls favor the leftist Smer-Social Democracy party, the senior member of the current coalition government, to top the voting with about 20%. Slovakia has 14 seats in the European Parliament.
The center-right German candidate to head the European Commission says he hopes voters will back a "Europe of stability" and a united and ambitious continent.
Manfred Weber, whose European People's Party group hopes to retain its status as the biggest in the European Parliament, said after voting in his native Bavaria Sunday: "I don't want to see a right-populist Europe (that) wants to destroy the idea of togetherness ... and I'm also against a Europe which is in the hands of the left."
Ska Keller, a German Green who heads her group's European election slate, said in Berlin that "the European Union should lead the way in climate protection. We need social cohesion, we need to strengthen democracy in Europe and I hope that this will meet with much support."
Austrian Chancellor Sebastian Kurz says he hopes the European Parliament elections will strengthen the center rather than parties on the far right and left.
Kurz told reporters in Vienna Sunday that he hopes his center-right People's Party will keep first place in the race for seats in the EU legislature.
The vote has turned into a first test of support ahead of a national election in September following the collapse of Kurz's governing coalition a week ago in a scandal surrounding the now-departed leader of the far-right Freedom Party, which was his junior coalition partner.
Regardless of the result, Kurz faces a no-confidence vote brought by the opposition in parliament Monday. He said he expects the Freedom Party and the Social Democrats to back it, which would bring him down.
Spanish caretaker Prime Minister Pedro Sánchez says that he hopes the outcome of Sunday's European and local elections will lead to more "political stability" for Spain as he starts his attempt to form a government.
Sánchez called on "all the political forces to open a horizon of political stability" after he voted early Sunday morning with his wife in Madrid.
He added that the elections are "to decide the future of progress and wellbeing for the entirety of our country and Europe."
Voter opinion polls point to a victory for Sánchez's Socialist Party in the European elections. Elections are also taking place for administrations in all Spain's cities, including deciding on a second term for the female mayors of Madrid and Barcelona, and 14 of its 19 regions.
Sánchez's Socialists won April 28 national elections in Spain, but fell short of winning an outright majority and will need to earn the support from rivals in Parliament to stay in power.
Hungary's prime minister says he hopes the European Parliament election will bring a shift toward political parties that want to stop migration.
Viktor Orban said Sunday after casting his vote at a school near his Budapest home that the issue of migration, which he believes is stoppable, "will reorganize the political spectrum in the European Union."
Orban, whose Fidesz party had its membership suspended in the center-right European People's Party, the largest political bloc in the EU parliament, because of concerns about Hungary's democracy, said Fidesz would want to stay in the EPP only if it can influence the group's future strategy.
Orban met recently with Italian Interior Minister Matteo Salvini, but has not committed to joining the more radically nationalist alliance that Salvini has been forming.
Fidesz is expected to win up to 14 of Hungary's 21 seats in the EU parliament.
Belgians are heading to the polls in European Union, national and regional elections Sunday.
Polls opened at 8 a.m. (0700GMT) and the first estimates and exit polls were expected by 6 p.m. (1700GMT). In the national elections Belgians are looking to end months of political limbo after the biggest party in the governing coalition quit over Prime Minister Charles Michel's support for the U.N. migration pact.
Michel has steered a caretaker government doing only day-to-day business since December, but with the country's 8 million voters choosing from more than a dozen parties, chances are that it will prove difficult to quickly form a stable coalition.
Bulgarians are voting in the European Parliament elections after a series of scandals overshadowed the debate on key issues of the EU's future.
Voters on Sunday are casting ballots for their country's 17 seats in the 751-member European Parliament. The vote is seen as a test for the center-right party of Prime Minister Boyko Borissov, which suffered a setback after senior officials were involved in corruption scandals.
Latest surveys show only three parties, belonging to mainstream European political groups, passing the election threshold — the ruling GERB party, the Socialist party, and the liberal MRF.
Projections suggest the nationalist and far-right vote will be split between several smaller parties, which could prevent them from capturing seats in the EU legislature.