What happens to the citizens of the country? What happens to the sovereignty of the country? What happens to the neighbors?
For example, what will happen if Greece becomes bankrupt?
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Nothing happens to a country's soveregnity legally.
That's because - unlike with most commercial lending (such as a mortgage), there is no tangible collateral in sovereign lending. In other words, the penalty for default isn't "we get to keep your collateral, or otherwise punish you" - there is no explicit direct penalty at all. Greece didn't borrow money by pledging its territory, or its citizens.
There are some legal avenues to the lenders (see the case against Argentine in US courts) but that doesn't seem to have a great deal of efficacy.
As the Economist article on the topic recounts it:
Spain’s 16th-century king, Philip II, reigned over four of his country’s defaults. Greece and Argentina have reneged on their commitments to bondholders seven and eight times respectively over the past 200 years. And most countries have defaulted at least once in their history.
Of course, as usual in international relations, the impact on soveregnity isn't always what's "legal", but what's "real" - in other words, if Germany had both the political will and the geopolitical power, they could very well have gone to war with Greece, and annexed territory and/or turned it into colony "until the debt is repaid". In other words, what's "legal" when it comes to soveregnity is basically what you have the world power to make stick, as Tibet found to its sadness when China made its occupation stick (Or Georgia and Ukrain, when Russia annexed parts of their territory, ditto with Germany and Japan couple of decades earlier).
What does happen realistically depends, but as lazarusL's answer covered in detail, basically your borrowing costs skyrocket as you are now a high credit risk, and you may have trouble attracting loans in the first place. Usually, nothing
That same Economist article goes on:
Defaults can also be very painful for the offending country, particularly if they are unexpected and disorderly. Domestic savers and investors, anticipating a fall in the value of the local currency, will scramble to withdraw their money from bank accounts and move it out of the country. To avoid bank-runs and precipitous currency depreciation, the government may shut down banks and impose capital controls. As punishment for default, capital markets will either impose punitive borrowing rates or refuse to lend at all. And credit-rating agencies will no doubt warn against investing in the country.
But as history shows, in most countries yield-hungry lenders will eventually start lending again so long as they are adequately rewarded for the risk they are taking on. Moreover, credit-default swaps—financial instruments which act as a form of insurance against sovereign and corporate defaults—allow bondholders to hedge their risk. But not all defaults are the same: Argentina defaulted again this year by refusing to pay $1.3 billion plus interest to the “holdouts” from 2001.
Critically, there is no international law or court for settling sovereign defaults, which helps explain why they are so varied in length and severity. More international regulation has been proposed—including powers to prevent minority holders from hijacking the process—but such conditions ultimately remain up to the issuing country. The first bond issuances since the new proposals (by Kazakhstan and Vietnam) include these clauses. Other countries might follow suit, but this doesn’t resolve the $900 billion of bonds outstanding that were issued under the old rules. Like any messy divorce, drawn out negotiations around defaults can be costly for all parties involved. Working towards better pre-nuptial terms might not be such a bad idea.
If a country "becomes bankrupt" what is actually happening is that it does not have the money to pay its debt. This will mean that it will default (not pay according to the terms) on some of its debt. This is very bad for the people who loaned the country money as they will not get their money back. The other effect is that future investors will be wary of loaning the country money, since the country has a demonstrated history of not paying its debt. This will make any future borrowing difficult and expensive (high interest rates). This will mean that the country may be unable to do deficit spending in times of recession as the costs of borrowing stimulus money will be very high. Additionally, if a country does not have the tax revenue base to support its current level of government spending, then it will have to abruptly reduce its level of government spending, for example by cutting government employee wages and stopping projects.
Many countries in history have fully or partially defaulted on their debt, and they usually maintain sovereignty.