What kind of situations do private equity funds invest in ?

Private equity funds intervene at every stage in a company's development, providing solutions to a range of different needs:

Venture Capital

Startup or very young companies with considerable development potential (ITC, biotech, etc.)

Development Capital

Companies needing capital for the next phase of their development (new production unit, international development, diversification, etc.).

Transmission Capital or LBO

Mature companies of all sizes where all or some of the shareholders wish to sell their shareholdings. 
An LBO is an opportunity for managers to acquire a stake in the company's capital: existing management (MBO), new management (MBI) or even a mixture of the two (BIMBO).

Turnaround Capital

Companies in difficulties.