Travel from Chicago to Graz . When is custom inspection?

Most airlines will check your bag all the way through to your final destination. Unless you have a connection that involves changing airports, such as flying into Heathrow and out of Gatwick. Basically, the first airport in Europe where you physically pick up your bags is where you'll go through customs.

That's not correct Christophe. Each airport handles passport control differently. For example, Heathrow and Schipol do NOT have passport checks for transfer passengers. On the other hand Brussels Airport does have passport checks for all transfer passengers. In fact, depending on what combination of airports you pass through, its entirely possible (but not typical) go go from the US to Europe without having you passport checked. This happened a couple of years ago when I flew from New York to Berlin via Schipol.

Passport control is ALWAYS at your first point of entry to the Schengen area. Customs control is usually at your final destination since your luggage is normally checked through all the way, and you can obviously not go through customs without your luggage. If your final leg is domestic, luggage is sometimes not checked through, but this is different from country to country and from airport to airport and has to do with the fact that at some airports, domestic flights arrive at a terminal without customs facilities, so you will clear customs as well as immigration at your transfer point. So, if you choose to transfer i Munich, you will deal with immigration in Munich and customs in Graz. If you transfer in Heathrow, you will clear immigration AND customs in Graz (UK is not part of Schengen). But probably, you will also go through checks in Heathrow as well, since this airport is full of checkpoints! Usually, it's immigration that takes time. At customs, most people are waived through.