In most cases, yes you can trade in another country's market. There are exceptions due to tax treaties or lack there of. Canada is the first country that comes to mind that will not allow US citizens to own shares of their companies directly. China is another.
You also have the ability to own ADR's, American Depository Receipts. These are receipts for shares of foreign companies held by US banks overseas. They trade just like actual shares on US exchanges.
Remember that when trading in foreign markets, you do not enjoy the disclosure protections offered by the Securities and Exchange disclosure that companies trading in the US must file. You will also be subject to changes in the value of the US dollar against the currencies of the country in which you own shares. A method to mitigate that problem is to own Exchange Traded Funds (ETF's) that own foreign shares and use defensive measures or hedge the currency risks between countries.
Best wishes in your international endeavors.
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