Strategies for Foreign Investing

Up to this point I’m most familiar with ADRs (American Depository Receipts). This seems to be the most direct way for a North American to invest in foreign stock. Are there others?

Let’s say I have EU citizenship. Can I open a bank account in the European country and invest with a local broker on a local exchange? This would be the most direct way to invest in the stock of other countries. Although I imagine it would be a tax headache to have several bank accounts in several countries for this purpose.

Obviously, I’m still fairly new to international investing, but these are some of the questions I’ll be investigating and posting on in the future. If you have other questions for me, let me know and I’ll see if I can find out about those too.