to have a good/bad reputation

to build/earn a reputation

reputation as something She soon established a reputation as a first-class cook.

reputation for something The company enjoys an international reputation for quality of design.

a politician with a reputation for honesty

reputation for doing something I'm aware of Mark's reputation for being late.

reputation of being something He has the reputation of being a hard worker.

to damage/tarnish/ruin somebody’s reputation

The weather in England is living up to its reputation (= is exactly as expected).

by reputation She is, by reputation, very difficult to please.