Refactoring a complex project live

We all faced situations when at first sight on the project we want to immediately bury it, get drunk and write a resignation letter. It's inevitable, we all face with old and bad code. How to work with it? How to make a low-quality code acceptable without breaking everything around? How to make it fast, well, and firmly?

Instead of theoretical advice let's take a big and desolated project, get horrified together, and fix it by discussing emerging problems and their solutions.