Currency is the risk with no equivalent at home. A US investor holding a foreign company owns two things at once: the shares, priced in their home currency, and that currency itself. The SEC states the consequence plainly: when the exchange rate between the dollar and the currency of an international investment changes, "it can increase or reduce your investment return." It adds a second, less familiar version of the problem, that "some countries may impose foreign currency controls that restrict or delay investors or the company invested in from moving currency out of a country." A rising currency can add to a poor year for the shares, and a falling one can erase a good one.
The SEC's other risks are worth reading as a list, because several are not financial at all. Its named risks are: access to different information, since many companies outside the US do not provide the same type of information as US public companies and it may not be available in English; the costs of international investing, which can be higher than investing in US companies; working with a broker or investment adviser, where it is "generally against the law for a broker, foreign or domestic, to contact a U.S. investor and solicit an investment unless the broker is registered with the SEC"; currency changes and currency controls; changes in market value, since all markets can experience dramatic changes; political, economic and social events, which are harder to assess from a distance; different levels of liquidity, since foreign markets may have lower volumes, fewer listed companies, shorter trading hours, and in some cases restrictions on what foreign investors may buy; legal remedies; and different market operations.
The legal remedies point is the one most often missed. The SEC warns that an investor with a problem "may not be able to seek certain legal remedies in U.S. courts as private plaintiffs," and that "even if they sue successfully in a U.S. court, they may not be able to collect on a U.S. judgment against a non-U.S. company," leaving them to rely on whatever remedies exist in the company's home country. That is a structural difference in what owning the share entitles you to, and no amount of diversification addresses it.
The five routes in, and how they differ. The SEC lists American Depositary Receipts, which is how the stocks of most non-US companies trade in US markets, with each ADR representing one or more shares of foreign stock or a fraction of a share and its price corresponding to the home-market price adjusted for that ratio. Then US-registered mutual funds, which the SEC notes are subject to US regulations protecting investors. Then US-registered ETFs, which offer similar benefits and trade through the day at fluctuating prices. Then US-listed foreign stocks, where a company lists directly here as well as at home. And finally trading on foreign markets through a US broker, where the SEC warns that such companies "are not likely to file reports with the SEC, so you will need to rely on other sources of information."
The fund-name distinction that changes what you own. The SEC separates four kinds of fund by name, and two of them are routinely treated as synonyms. Global funds "invest primarily in foreign companies, but may also invest in U.S. companies." International funds "generally limit their investments to companies outside the U.S." Regional or country funds invest principally in a particular region or a single country. International index funds seek to track a particular foreign or international market index. So a global fund held alongside a US total-market fund can leave a portfolio holding more US equity than intended, and an international fund will not.
Fund names in this territory are also governed by the SEC's fund names rule, which treats a name suggesting a focus on a particular country or geographic region as requiring a policy to invest at least 80% of assets accordingly. The geographic test is that the investments be "tied economically to the particular country or geographic region suggested by its name," which is deliberately broader than where a company is incorporated.
How much to hold is an allocation question, not a definitional one. The range of defensible answers is wide, running from holding non-US stocks at their share of world market value down to holding none, and the tendency to hold far more of one's home market than its size would suggest is common enough to have its own name in home country bias.