So quite simple, as the title states: as a UK national, can I buy shares in a US based company?
A little more than this however, is do I have to sign up to a US brokerage firm, or can I do this from a UK based brokerage firm?
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I am assuming for simplicity's sake that you mean a US-traded, as well as US-based, company*.
Which shares exactly, this depends on your broker. But the answer is yes it's possible with a UK brokerage firm.
To take a mainstream example, with Hargreaves Lansdown for example, most US shares from the main indexes (NYSE, Nasdaq) are available.
* because of course, if you are referring to a US-based private (unlisted) company, or a US-based but (say) Buenos Aires-listed company, the answer is more complicated.
Generally, you can buy shares from every company which trades at any of the exchanges your broker works with.
E. g., if Tesla is traded in Paris, Frankfurt, but not in London, you cannot buy it if your broker only works with the London Exchange.
So whether it is possible or not depends on the company whose shares you are interested in and the exchanges your broker works with.
As you are based in the UK, I suppose your broker works at the LSE. There you can buy Tesla shares: https://www.londonstockexchange.com/exchange/prices-and-markets/stocks/summary/company-summary/US88160R1014GBUSDGES1.html?lang=en