How do I invest in retirement if I'm planning to leave the US?

The purpose of American retirement plans are to stimulate savings by deferring US income taxes now and paying them later.

Since in all likelihood you won't owe US income taxes when you're abroad, there's no purpose to deposit money in an IRA. (401k accounts are irrelevant, since the deposits are handled by your employer, and your Upper Elbonian employer won't expend the effort to run a 401k plan just for you, when you don't owe any US taxes anyway.)

Besides, most countries don't have the concept of the IRA or 401(k) because they have higher government pensions.

This is a very big subject and, unfortunately, there are no universal answers. The general advice though is to take advantage of the retirement system where you are resident when you are resident there. If you have sizeable retirement investments, you should consult an accountant knowledgeable in both tax systems to help you with tax planning before you move to a new country.

Foreign Retirement Investments and Savings

The details will always depend on your citizenship, where you are a tax resident, and any tax treaties between your country of citizenship and the country where you are a tax resident so no specific advice can be given.

As a US citizen, if you choose to invest in foreign vehicles for retirement while living abroad:

1. Check the relevant tax treaty and consult a knowledgeable tax professional to understand how pensions or retirement savings options like ISAs will be treated from your US tax perspective. Especially make sure that you do not choose an option that is considered a PFIC unless you have no other choice and/or you know what you're doing and choose to do this.
2. Familiarize yourself with the tax implications of mortgages denominated in foreign currency before investing in real estate. Specifically, make sure you understand whether paying off the mortgage may result in ordinary gains based on currency exchange rate fluctuations.

Should you choose to return to the US, the relevant tax treaty will also explain how your foreign retirement vehicles will be treated in the US.

US-based Retirement Investments and Savings

In terms of taking advantage of your US-based retirement vehicles:

If there is a tax treaty in place, you may be able to benefit from and possibly contribute to your US-based retirement vehicles. If there is not a tax treaty in place or it doesn't cover the specific vehicle you've used (e.g. Roth IRA), you'll need to check with a tax professional knowledgeable about both tax systems.

When the US has a tax treaty with a country, it generally tries to get the other country to apply similar tax treatment to US-based retirement vehicles (401(k), Traditional IRA, Roth IRA, etc) as the US does. However, not all tax treaties achieve this and some that do are out of date and don't cover things like Roth accounts. I'm not sure if any cover things like HSAs or 529s.

If your US-based retirement vehicle is covered by a US tax treaty, you can probably expect to withdraw money from it in retirement under similar rules as you would if you were resident in the US. For Traditional IRAs and 401(k)s, this could mean distributions are taxed as income (at your country of residence's income tax rates). For Roth IRAs, this could mean distributions are tax free.

In some situations you may also be able to continue contributing to US-based retirement vehicles from your country of residence and still enjoy the expected tax advantages. For example, you may be able to treat contributions to a traditional IRA as tax-deductible contributions to a "pension". This gets complicated though when considering the Foreign Earned Income Exclusion (FEIE) as income excluded by the FEIE is not eligible for contribution to an IRA.