Many threads on this board are started by people wondering whether they should move here or not. I thought it might be useful to group together some of big considerations, though obviously everyone's individual circumstances and preferences will differ. Feel free to amend/add thoughts!
The city of Cambridge is around 130k inhabitants. It punches above its weight in terms of reputation, education and cultural activities, but basically it's the size of Worcester or Bedford. If you think of it as a large town rather than a true city, that will give you a better flavour of what it's like.
It is relatively isolated and is smaller than the other regional cities in East Anglia - Norwich and Peterborough (both 180k inhabitants); and Ipswich (160k). All of these cities are about an hour's drive away, the same time it takes to get to Kings Cross on the train.
It is has a very international population, but it's also a very transient population - not just students, but also workers and families. That has an effect on how you might experience community here.
The picture-book images of Cambridge apply to a 1km area in the centre of the city. You won't be living there! City housing is typically either Victorian terraces, C20th suburbs or new build. Demand for housing anywhere in the city massively outstrips supply, so expect to pay a lot for what you get!
It is experiencing severe growing pains - too many cars, especially inbound commuting, and poor public transport options. If you are interested in living in the city, think about whether you will be willing to walk or cycle to keep mobile.
It has an increasingly complicated relationship with the villages surrounding it. Rural residents like their rural surroundings and want to keep them that way; but also want to be able to access Cambridge easily for employment, shopping, schools, leisure, etc. There are big discussions ongoing about how this can be achieved without trashing the city - building mass transit schemes across the Green Belt, congestion charging, lots of ideas, but no clarity yet on solutions. Plan your journey to work carefully!
On the plus side, there are no 'bad' areas and the schools are generally good.