When the company I work in hired new managers, they offered us to overview someone's code on every meeting. We have meetings every two weeks, so each time one of developers was to show his/her code on the projector, and others were going to discuss it.

I thought this would be great: each developer would be more careful when writing code and we could share our experience better. But somehow we forgot about this and the offer just remained an offer.

What are the benefits of this and are there any drawbacks?

Code reviews are a great practice.

It is probably the best way to learn from mistakes and to see how certain problems are solved by others. It is also one of the best way to maintain quality in a code base.

Code reviews happen in many companies, though it is difficult to say that there is a specific process that they all follow. 

In more formal code review, a senior (or several seniors) will sit together with a developer to review their code, offering suggestions and teaching at the same time.

Additional benefits to code reviews (as commented on for this question), include:


A great way to teach and learn
They are one of the best ways to improve and keep the consistency of a code base (style and idioms)
They help ensure all team members understand the style and idioms used in the project and how to use them
Code reviews will speed up development as they catch bugs and design flaws early (so, though they may slow down initial development, they pay dividends in later development cycles)
There is tooling support that helps streamline the code review process

