How do I know when my S-Corp has to pay taxes, and what do I need to fill out?

You need to speak with an accountant. Or at least hire a bookkeeper Unfortunately, it's another expense, but a needed one.

I am not an accountant and this is not "professional advice."

I do have an S-Corp, though.

You need a corporate accountant. A public accountant cannot do corporate taxes.

You pay by the March 15 deadline like everyone else. UNLESS you are earning more than a certain amount a year, which your account can advise you of--then you will likely pay quarterly remittance which is an estimate based on revenue and then adjusted at fiscal year-end. If you're earning under say $200K, don't worry about it: you're still on the standard tax plan.

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In short - you must file as often as required per your state plus the federal. If you have to claim $0 revenue, net losses - do so. If anything it benefits you, as investor. Go talk to a cpa for further advice, also with s corps make sure to record minutes for any changes or efforts made on the companies behalf. Sometimes they are needed, sometimes not.
Good luck, and talk to a cpa directly.

I have an S-Corp and several other corps, but I'm not an accountant.

Definitely hire a qualified accountant, as the various state and federal dates and taxes can be confusing. You can also use a Registered Agent service that offer notifications and managed reports so that you don't miss dates. Companies like MyLLC.com (I am not endorsing this - just an example)

S corporations are flow through entities for U.S. federal and state income tax purposes. So, the entity itself is not subject to income taxes. The net earnings will flow through to your Form 1040 and you pay federal and state taxes at the shareholder level. S corporations are required to file Form 1120-S on an annual basis. Each shareholder in the S corporation is provided a Schedule K-1 which shows their allocable share of income and cash distributions during the year. The S corporation returns are due by March 15th, but you may file for an extension if you need to more time to gather information and complete the return.

The big benefit of S-corp taxation is that S-corporation shareholders do not have to pay self-employment tax on their share of the business’s profits. Her revenues from the business are $60,000 per year, and her annual expenses total $10,000. Therefore, her S-corp’s profit for the year is $50,000. This exact question is frequently the topic of debate in court cases between the IRS and business owners who are, allegedly, paying themselves an unreasonably small salary to save on self-employment taxes. A shareholder’s cost basis in an S-corporation is increased by his allocated share of the business’s income and by contributions he makes to the business. His basis will be decreased by his share of the business’s losses and by distributions he receives from the business. In the calendar year in which the business is formed, the business pays Austin a salary of $30,000, after which it has remaining ordinary business income of $20,000. The $30,000 salary will be taxable to Austin as ordinary income, and it will be subject to normal payroll taxes as well. The $20,000 ordinary business income will be taxable to Austin as ordinary income, but it will not be subject to payroll taxes or self-employment tax. Of note, any compensation that the S-corporation pays to you is not considered to be pass-through income. It is only allocations of profit from the S-corporation that are pass-through income.
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An S-Corp must pay taxes on all income and should file their taxes annually by completing IRS Form 1120S.

As an S-Corporation, your business entity itself generally does not pay income tax at the federal level. Instead, the income and losses of the S-Corporation are passed through to the shareholders, who report their share of the income or loss on their individual tax returns. However, there are still some tax-related responsibilities that your S-Corporation will have.

Here are some key tax-related dates and forms to be aware of:

March 15: S-Corporations must file their annual tax return (Form 1120S) by March 15 of each year. This return reports the S-Corporation's income and expenses for the previous year, and provides each shareholder with a Schedule K-1, which reports their share of the income or loss.

Quarterly estimated tax payments: If the S-Corporation has taxable income, it may need to make estimated tax payments on a quarterly basis throughout the year using Form 1120-W. These payments are generally due on the 15th day of April, June, September, and December.

State and local taxes: In addition to federal taxes, your S-Corporation may be subject to state and local taxes, which vary depending on your location and the specific tax laws in your area. You may need to file additional tax forms and pay additional taxes at the state and local level.

It's important to note that tax laws and regulations can be complex, and it's often a good idea to work with a qualified tax professional or accountant to ensure that your S-Corporation is meeting all of its tax obligations. They can help you determine when your S-Corporation needs to pay taxes, what forms you need to fill out, and how much you need to pay.

As an S-Corp in the US, your tax obligations and filing requirements are slightly different from those of a C-Corp. Here's an overview of when and how an S-Corp pays taxes and the forms you may need to fill out:
1) If your S-Corp is expected to have a tax liability of $500 or more for the year, you may need to make estimated tax payments on behalf of the shareholders.
2)As an S-Corp, your company itself does not pay federal income taxes. The shareholders are then responsible for paying taxes on their share of the S-Corp income.
3)Despite not paying federal income taxes directly, an S-Corp is still required to file an annual tax return using Form 1120S, U.S. Income Tax Return for an S Corporation.

As per my knowledge I can that in almost all countries of world, the CAs and Taxation Authority will ask you only one thing that from where you get the amount for expanses as you do not have any sales ? You just have to clarify that in a very proper way and for that you take the help of your accountant or revenue lawyer. They can guide you well as per your countries laws.

And still if you want any help then I am there to help you.