One may think that the European Union is a strong, self-sustained, powerful and fault free Union. But the reality states a worse situation, the EU is in a strong crisis ever since the EU Constitution was rejected and until now they are struggling to turn down the tension. But aside of that constitutional or let me say existential problem they are also facing a lot of problems within themselves and with other European countries - non members or candidate countries.
Belgium (EU member) for example lately ran away from the possibility of partitioning its land between the French speaking community and the Dutch speaking community which may either lead to gaining independence or respective accession to either France or the Netherlands. After the June 2007 Elections they went into deep crisis spending a long time constituting their new Government. In March 2008 they managed to create a fragile government that many still wonder how long it will last.
Slovenia (EU member) has a major dispute with Croatia (EU candidate) over marking the border between them. Ever since their break up from the former Yugoslavia they haven't yet reach an agreement over that issue although Slovenia has several times warned their neighbour that this issue may jeopardize the EU candidacy.
Cyprus (EU member) is the only country in EU that has territories under Turkish occupation (as they consider) after the Turkish invasion in the 1970s. Although the United Nations tried to solve the issue before Cyprus' accession to the EU in 2004 with a referendum but the Greek population in Cyprus rejected the re-unification of the Mediterranean island whereas the Turkish population agreed. Whats more interesting is how Greece is meddling in between Cyprus and Turkey and Greece has warned Turkey several times that they will face a Greek Veto if Turkey doesn't withdraw its troops from northen Cyprus.
Greek-Turkish Dispute also referred to as Aegean dispute. The term Aegean dispute refers to a set of interrelated controversial issues between Greece and Turkey over sovereignty and related rights in the area of the Aegean Sea. This set of conflicts has had a large effect on the relations between the two countries since the 1970s. This led twice to crises coming close to the outbreak of military hostilities, in 1987 and in early 1996. And the dispute is also jeopardizing Turkey's aspirations to gain EU full membership.
Greenland which is until now under Danish rule but is granted home rule authority is seeking a way to declare independence since it is not part of EU and that will lead to a great loss for the Danish Economy.
Gibraltar, Spanish territory conquered by Britain in the 1700s. Spain attempted to recapture the territory militarily, with a number of failed sieges, and reclamation of the territory by peaceful means remains Government policy until today.
Greek- Macedonian naming dispute over the name of the Republic of Macedonia. Greece is still opposing the right of a nation of self-identification and is absurdly trying to impose a name to another nation just to cover Greece's massacre over ethnic Macedonians who were living in Greece during the Greek civil war (1950s). They consider that by denying the existence of a Macedonian state they can deny the existence of a Macedonian minority within its border. Greece's recently VETO-ed Macedonia's candidacy to NATO just because they don't like the name. Negotiations are still occurring under the patronage of the UN.
South Ossitia. (non EU member) is a self proclaimed Republic between Russian and Georgian Border that is still seeking UN recognition. It has been used by Russia as Trojan Horse in Georgia. The later still considers South Ossitia to be part of its territory and the dispute is still waiting for a solution. Recently Georgia expressed their intention to Join the NATO alliance to which Russia strongly opposed considering it a threat to Russia's Territorial integrity and sovereignty.
Bosnia and Hercegovina. Country consisted of two entities : Republika Srpska and the Federation of Bosnia and Hercegovina. The entities, based largely on the territories held by the two warring sides at the time, were formally established by the Dayton peace agreement in 1995 due to the tremendous changes in Bosnia and Herzegovina's ethnic structure. It is unclear till now till when this fragile union will last having three different ethnicities (Serbian , Croatian and Bosnian) and it is a possible war source.
Kosovo is another self-proclaimed defacto independent State, although Serbia still considers it part of its territory. So far about 40 something country recognized its independence but the new born country has a lot of to fight in order to get full UN recognition since they are facing a strong opposition from Russia who is Serbia's strongest Ally. Still waiting for a solution that also may reach a critical war level.