From the outside, it could be a Premier League football ground on a day the team is playing away.

There is the same sense of enclosed space behind featureless walls, with a vast paved area in the foreground for surging crowds chanting inanities.

There are even flags flapping in the breeze atop vertiginously tall poles, although none sport the colours of any football club known to me.

And what are those twin towers, 21 storeys high and gleaming gold in the Luxembourg sun?

It is, of course, the Court of Justice of the European Union (CJEU), or what we used to call the European Court of Justice – or ECJ, an institution which was discussed at last week's Council of Bars and Law Societies of Europe.

It’s the highest court in the EU.

The building is not that hideous, really, and certainly not in the context of the Kirchberg plateau, the area of the city of Luxembourg resembling a high-maintenance industrial estate where international law firms, banks, the Luxembourg Chamber of Commerce and the CJEU ply their trade.

The court is expanding, too, with a third gold tower planned and building work almost everywhere you look.

But before we begin the guided tour proper, let’s review what we know about the CJEU.

It was set up in 1951 to implement the legal framework of the European Coal & Steel Community (the what?), but has since developed beyond recognition to become the court that enforces EU law throughout the 27 member states and even outranks national supreme courts.

Its judgments can affect both the member states of the EU and individual citizens of the EU, and it is the referee between member states, institutions and individuals in disputes relating to EU law.

There are 27 judges, one from each member state, and each has his or her own courtroom.

The judges are supported by eight advocates-general who deliver legal opinions on cases.

The workload of the court has increased from 79 cases in 1970 to 1,890 in 2009, of which it completed 588.

It is appointing more advocates-general to help clear the backlog.

In 2009, the court passed 377 judgments against member states for failing to fulfil their obligations, including nine against the UK.

And the court employs some 2,000 people, of whom around half work in the translation department.

Now back to the guided tour.

Once through the tinted glass doors, you enter a structure built from a giant’s Meccano set.

There are exposed metal beams, see-through lifts, sweeping staircases like flumes in a children’s swimming pool.

One courtroom we visited smelt of wood polish and was flanked by two-tiers of translators’ booths, the translators themselves shielded from view by black tinted glass.

Another had a ceiling made of netting, as though a daring young man on a flying trapeze was soon to fly through the air with the greatest of ease.

The entrance to one court was a short passageway ‘wall-papered’ with tin mesh – mined in Luxembourg and fashioned in France, our guide told us – and illuminated by lamps like those used by miners.

Mining is an important industry in Luxembourg, we were told.

Fixtures and fittings were all very modern: earphones stored in the arms of your chair, the biggest library of EU law books in the world.

But perhaps best of all was the somewhat spookily named Deliberation Chamber.

This comprised a long table with 27 seats for 27 judges, all labelled with their names and set out with law books, pencils and bottles of water.

The judges meet here to deliberate every Tuesday evening, sometimes late into the night.

The shelves lining the room floor to ceiling are ominously empty.

The court’s art collection also sticks in the memory.

There is one particular sculpture set into the ground outside the court that represents growth, the guide informed us, but looks like the writhing legs of two people who have nosedived into quicksand – and regretted it.

Other pieces were inspired by the art of Amazonian Indians, and another looked like a big fat fish – and may have been, for all I know.

Was the tour enlightening?

Yes, most certainly. It’s always humbling to visit a great institution and anyway, my flippancy notwithstanding, the building itself is likely to assume a new grandeur and gravitas for generations to come.

But I’m still not sure about the tin mesh wall lining – does tin go rusty?