The University of Coimbra is the latest addition to the Portuguese World Heritage List and one of the few sites in the category secular education and science. The inscribed area comprises buildings from the 12th century (the old cathedral) to the 20th century. You have a nice view from the opposite bank of the Mondego river up to the hill with the main university buildings towering above the city.

I entered the upper town (Alta) through the Arco de Almedina. The way up to the university buildings lead past the Sé Velha, that looks more like a castle than a cathedral. It is worth a visit, in particular the cloister. I was surprised to find such a quite large cloister between these steep and winding streets. However, after having visited Alcobaça and Batalha, I was not that impressed anymore by churches and cloisters. But if you make the trip in the opposite direction, you can take it as an introduction to the World Heritage monasteries in central Portugal.
The main sight in Coimbra is the former Royal Palace of Alcaçova at the highest point of the city. The historic buildings are lined around the inner courtyard: Biblioteca Joanina, Capela de São Miguel, the bell tower Torre da Universidade, the Palace with the Sala Grande and a few other historical rooms. Tickets are available at the new library just across the street, and you have to book a time slot for the Joanine Library. The visit to the Biblioteca Joanina takes about 20 minutes, taking photos is not allowed (but in the other buildings). The library is a great example of Baroque architecture, but the other buildings are a bit underwhelming. I liked best the beautiful courtyard, the ensemble of buildings and the open south side with a great view over the Rio Mondego. And do not miss the seal of the University in front of the main gate, a fine example of Calçada Portuguesa (Portuguese pavement).
Next to the historic buildings are the most recent buildings in the core zone: several faculties built in the 1940s, the period of the "Estado Novo" dictatorship. The photo shows the two architectural styles, seen from the Baroque bell-tower of the Alcaçova Palace.
Actually, the best part of the visit was to stroll through the streets of the university city and explore the buildings that are still in use today. Most of the Colleges were open, at least the courtyard, the corridors and the cafeteria were accessible. Examples are the College of S. Jeronimo and the Royal College of Arts with a nice courtyard (both at Alta), the College of S. Agostinho (halfway down to Sofia), and the College de Carmo (at Sofia). I would recommend visiting Coimbra on a weekday and not during semester break, to get also a glimpse of the university life.

Coimbra is also known for its special style of Fado, which is different from the Lisbon version, less sad but more lyrical and romantic. The Fado de Coimbra is sung only by men because it is a student tradition that originated at a time when only men were admitted to the university. Though I spent only one evening in Coimbra, I attended two Fado concerts. First, at the Fado ao Centro, which at first glance looks like a tourist trap, but actually it is a cultural center to promote Fado. They also explained the history and tradition of the music, I can highly recommend the show. And later in the evening, rather by chance, I had a second concert at the Café Santa Cruz, a nice traditional coffee bar next to the Santa Cruz monastery.

The historic ensemble of the Royal Palace alone would not justify inscription on the WH list in my opinion. But Coimbra is an example of a university city, testimony of the interaction of a university and a city over seven centuries. However, as a university city it can not compete with Oxford and Cambridge. I spent two days in Oxford in the previous year (in May 2017) and I visited Cambridge 15 years ago, both are great sites. And so I left Coimbra with exactly the same question that I asked myself at Oxford: why the hell is this (Oxford) not a World Heritage Site?