Sources: the official FAQ and blog posts.
See also the official privileges page for bounties.
What is a bounty?
A bounty is a special reputation award given to answers. This feature was designed to motivate answerers, and help questions get the answers they deserve.
Bounty awards are funded by the personal reputation of the users who offer them. The award can be anywhere from 50 to 500 reputation, in 50 point increments. Note: If you are offering a bounty on a question that you have already posted an answer to, your minimum spend is 100 reputation (not 50).
While you do not need to be the owner of a question to start a bounty on that question, only one bounty can be active on a question at once, and each user can only have up to three active bounties at once.
As of June 2010, the bounty system is decoupled from accepting an answer.
What is the "Featured" tab on the homepage?
How can I search for questions that have a bounty attached?
Questions with an active bounty appear with a special icon in all question lists, and are also visible on the home page's Featured tab. Questions in that tab are sorted by the amount of time left on their bounties. The closer a bounty is to expiring, the higher it will be on the list.
How do I start a bounty? When can I start a bounty?
A bounty can be started on a question two days after the question was asked.
To start a bounty, click on the start a bounty link at the bottom of an eligible question. The bounty panel will open. Use the dropdown to allocate the bounty amount -- anywhere between 50 and 500 reputation, in 50 point increments.
You must have at least 75 reputation to start a bounty, and at least as much reputation as the bounty amount.
The bounty award will be subtracted from your reputation when the bounty is started, not when it is awarded.
Can I use Markdown formatting in the bounty remarks? Can I edit the remark?
Bounty notices support the markdown mini format using used for comments. The following are allowed:
_italic_ and **bold** text,
inline `code in backticks`,
and [basic links](http://example.com).
You cannot edit the remark, so validate carefully before saving.
How long is the bounty period?
1 week = 7 days.
As of June 2011, bounty offerers may award bounties up to 24 hours after the bounty period ends (known as the grace period), to allow them to evaluate answers posted at the end of the bounty period. During the grace period, there will be no indication in the questions lists (e.g. front page, featured tab) that the question has a bounty, as confirmed here.
Note that even if you don't have an answer, or only a self-answer, the grace period is still in effect for 24 hours. Answers posted during this time period may be awarded the bounty.
How do I award a bounty?
A bounty can be awarded 24 hours after the bounty was started. Any answer to the question, even an existing answer that has already been awarded one or more bounties, is eligible to be manually awarded the bounty. This still applies if you have already awarded a former bounty to this answer yourself.
To award the bounty manually, click on the +50 (or whatever bounty amount was allocated) button on the left side of the answer you want to award:

Can I award a bounty to my own answer?
No. This used to be possible, but it has been disabled. The user would not get the reputation back, and the bounty will be displayed as +0, “this answer has been awarded bounty worth 0 reputation”.
Can I award a bounty to an old answer?
Yes, you can award your bounty to any answer on the question. This makes it possible for users to reward particularly good answers with more rep than a standard upvote would provide.
To indicate that your bounty will be awarded to an existing answer, choose "Reward existing answer" when asked "Why are you starting this bounty?"

Keep in mind, that the bounty can be awarded only after a minimum of 24 hours, after starting the bounty.
If I offer a bounty on someone else's question, can I award the bounty to their answer?
Yes, a bounty can be awarded to any answer except one posted by the person offering the bounty.
This means that if you offer a bounty on someone else's question, and the original poster of that question posts an answer, you can award the bounty to their answer.
After awarding the bounty, can I remove it or move it to another answer at a later time?
No, awarding is permanent. (But you're warned about that when actually awarding the bounty.)
What happens if there's no answer after the bounty period?
If after the end of the bounty period a question has no answers, no bounty will be awarded and the question will no longer be featured.
Bounties are best understood as exchanging reputation for higher question visibility and increased answerer motivation. A bounty does not guarantee a response and is not refunded if none are received.
What happens if I feel my question is still unanswered?
What is automatic awarding?
Approximately 24 hours after the end of the bounty period, if the bounty starter has not manually awarded the bounty, the bounty may be awarded automatically.
If the bounty starter accepted an answer during the bounty period, that answer is awarded the bounty (provided that the answer was also posted during the bounty period). Answers created or accepted before the bounty period are not eligible to be awarded the bounty automatically.
Otherwise, the bounty is awarded to the highest-scored answer out of those which...
- ...were posted after the bounty was started
- ...have a score of at least +2
- ...were not written by the bounty starter
If two or more eligible answers have the same score (if their scores are tied), the oldest answer is awarded the bounty.
If none of these conditions apply, the bounty is not awarded to any answer, and is not refunded to the bounty starter.
Automatically-awarded bounties are half of the amount originally offered, unless the answer is accepted - accepted answers always receive the full bounty.
Are bounty awards exempt from the 200 points/day reputation limit?
Yes, bounty awards are exempt from the daily reputation cap.
How does Community Wiki mode affect bounties?
Bounties are not affected by community wiki mode. When you award a bounty to an answer marked community wiki, the reputation bonus will be awarded to the user who posted the original revision of the answer.
Can I offer a second bounty after the first one has expired?
Can I raise my bounty?
You can offer as many bounties on a question as you want. However, only one bounty can be active on a question at a time. Moreover, any user may have at most 3 concurrent bounties at a time.
Note that if you offer several bounties on the same question, you will have to double the amount each time (or more). That is, if your first bounty was worth 50 reputation, your second bounty on the same question will have to be for at least 100, your third for at least 200 and so on. If you've already offered a bounty for more than 250, you can still offer more bounties for 500 (the maximum amount) as long as you like (or as long as you have the rep). This doubling applies only to bounties by the same user on the same question.
What happens if a bounty question is closed, or deleted?
Why can't I vote to close or migrate a bounty question?
Bounty questions cannot be closed directly.
However, diamond moderators can refund bounties, which would then allow it to be closed, migrated, or deleted like any other question. (source)
What happens if I delete my account while I have an active bounty?
The bounty will just be moved to (owned) by Community and will be awarded at the end of the period as they normally would. (source)
Do I keep an awarded bounty when I delete my answer?
When an answer that received a bounty is deleted, all reputation including the bounty are revoked. (This may take 5 minutes to be visible.) The bounty is not given back to the user who awarded it.
Do I get the reputation back when question I placed bounty that was already awarded is being deleted or migrated?
Yes. The change is not immediate like other reputation changes but the reputation is given back (Source – marked status-bydesign is official enough).
If the question is undeleted or migration is rejected, the reputation is reduced again.
Can I cancel my bounty?
No. Once you start a bounty, you cannot cancel it.
If you feel there are exceptional circumstances, flag the question for moderator attention to explain the situation.
Can I affect / divide part of my bounty to two (or more) users?
No. If multiple answers to a single question have earned bounties, it's because the question has been subject to multiple bounties.
How many outstanding bounties can a single user have?
Three. You must award a bounty after this point to start a new bounty.