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To enter Australia for up to 90 days you need a valid passport and a visa (New Zealand nationals are the exception). These days, instead of a visa label or stamp in your passport, citizens of the United States (and many other countries) can get an Electronic Travel Authority (ETA). This is an electronically stored travel permit. It saves you time both when you apply—the process is all online—and when you arrive in Australia.
To obtain an ETA for Australia you must: 1) hold an ETA-eligible passport; 2) be visiting Australia for tourism, family, or business; 3) stay less than three months; 4) be in good health; and 5) have no criminal convictions. The Visitor ETA allows you as many visits of up to 90 days as you like within a 12-month period, but remember that no work in the country is allowed. If you're visiting Australia on business, a Short Validity Business ETA might be more appropriate. Technically, both are free of charge, but you need to pay a A$20 service charge by credit card. Children traveling on a parent's passport also need an ETA. You can apply for the ETA yourself or your travel agent can do it for you.
If you don't meet the ETA requirements or need a different kind of visa, you should contact your nearest Australian diplomatic office well in advance of your trip, as processing other visas takes time. Equally, if you plan to stay longer than three months you must obtain a paper visa (there's a A$130 fee). If you travel to Australia on an under-three-month ETA and later decide to extend your visit, then you must apply for a visa at the nearest Australian Immigration regional office (there's a A$250 fee).
At present, Australia doesn't require a notarized letter of permission if only one parent is traveling with a child, but it's always best to err on the side of caution and take along such a letter if you can.