(assuming that we're talking about the AWS serverless functions) Like everything else, it's situational.

Upsides of lambdas are ease of deployment (no need to worry about servers, that's kind of the whole point of serverless), virtually infinite horizontal scaling, and a very generous free tier.

Downsides are relatively slow cold starts, difficulty of exposing to the outside world (eg via HTTP route), and lack of state management.

Personally I like using Lambda as glue between different parts of the AWS ecosystem, or to handle events, dispatch notifications etc.

However I would definitely not use Lambda for anything remotely resembling a stateful web app, for instance. The slow cold starts and inherent statelessness are going to make that difficult. Also API Gateway is a huge pain to work with, or was last time I looked at it.