What are some achievement of Europe in the Middle Ages?

A lot happened in the middle ages.

At the beginning of the middle ages, Western Europe was in a shambles, with wave after wave of Germanic tribes dismembering the old Roman Empire of the West. These Germanic tribes gave rise to kingdoms that looked back to Roman Law and Order with some interest once they regarded their land as their own possession and began to want to improve it. The result was a rise of centralized government, which was feudal in nature, but which had a vested interest in prosperity.

Charlemagne lead the Franks into a period of rising literacy, importing the best teachers into his court. Alfred the Great of England did the same, going a step further and making sure education was to be done in English. He had great books translated for the purpose and said he wanted all free men of England taught to read if it was possible to do so.

Schools were opening in the most surprising places and times. The first primary school was opened in the fifth century, several church schools in England date from the seventh and eighth centuries, a state run secular school, the Beverley Grammar School, was opened in Northumbria in 700 AD (now in Yorkshire). By the end of the Viking Age, there were schools opened in Denmark, Iceland, and Sweden (I was astonished to find this out).

The first universities opened in the eleventh and twelfth centuries, and there were more than seventy opened during the middle ages, altogether.

There were a lot of inventions in the middle ages. The introduction of the horse collar made agriculture so much more efficient that the population of Europe at the end of the middle ages was greater than it had been during the height of the Roman empire. Stirrups, new looms, rag content paper, gunpowder, canons, the printing press, and movable type were all products of the middle ages.

Feudalism, which was characteristic of the middle ages, gradually gave way during the period. By the end of the middle ages it was pretty much gone in Scotland, England and Wales.

At the same time, the banking system rose in the middle ages, first influenced by the Knights Templar, and, when they were destroyed, it was taken over by such groups of people as the powerful Medici family, which was mercantile.

And in northern Europe, the Hansas formed the Hanseatic League, a mercantile organization that rivaled nations for power and had its own navy.

Art and architecture began to rise during the Early Middle Ages, and brought the about the Gothic period, with its stained glass and lofty cathedrals. This, in turn, gave way to new styles of art that presaged the Renaissance.