1. make a sudden advance, overcome an obstacle
have a breakthrough in talks/negotiations/your career
A 'breakthrough' is a sudden development or discovery of importance. You can 'have a breakthrough' or 'make a breakthrough', which means to have been involved in the process of a breakthrough. This is a term used often to talk about scientific discoveries and significant research, but can be used to talk about more domestic issues also: "I made a breakthrough with my children today when one of them washed the dishes without being asked." Similar phrases include 'make a finding' and 'a major development.