Accepted means your tax return is now in the government's hands and has passed the initial inspection (your verification info is correct, dependents haven't already been claimed by someone else, etc.).
After acceptance, the next step is for the government to approve your refund.
First, they look for things like back taxes and unpaid child support. If they find any debts, they'll offset (reduce) your refund to cover the outstanding amount. Once they are satisfied that you have no outstanding debts, they will approve and then issue your refund.
Think of the e-filing/refund process along the lines of applying for a job; although the employer (government tax agency) has received (accepted) your job application, they still have to review (approve) your application before they grant you an interview (refund).
The IRS says 9 out of 10 filers who efile and choose direct deposit will get their refunds within 21 days. You can monitor it with the IRS tool mentioned below. Some returns take longer for additional processing due to specific items in the return. Some are pulled at random for more thorough investigation.
Also, if you have EIC (earned income credit, aka EITC) or ACTC (additional child tax credit), the IRS will not start releasing those refunds until February 15. That means people with those types of credits may not start receiving their refunds until around February 27.
NOTE: If you chose to pay any TurboTax fees out of your Federal refund, that is another link in the chain, since your funds go first from the IRS to an intermediary bank where the fees are subtracted. Then that intermediary bank sends the rest of the funds to your bank in a second direct deposit.
You can monitor the status of your Federal refund at this IRS page below. If you efiled, wait 24 hours after acceptance before trying this tool, or it may not find it.
To track a state refund:
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