Adoption tax credit
According to the IRS, "Tax benefits for adoption include both a tax credit for qualified adoption expenses paid to adopt an eligible child and an exclusion for employer-provided adoption assistance. For tax years 1997 through 2009, the credit was nonrefundable. For 2010 and 2011, the credit was refundable. For tax year 2012, the credit has reverted to being nonrefundable, with a maximum amount (dollar limitation) of $12,650 per child."
The American Taxpayer Relief Act of 2012 enacted January 2, 2012 permanently extended the adoption tax credit.. For 2013 the nonrefundable maximum tax credit per child is $12,970. The credit begins to phase out when modified adjusted gross income (MAGI) exceeds $194,580 and is eliminated when MAGI exceeds $234,580. The tax credit is claimed on IRS form 8839 Qualified Adoption Expenses. 
Tax Year to Claim the Adoption Tax Credit
Domestic adoptions expenses are claimed for a tax credit in the tax year following when they are paid. Both domestic and foreign adoption expenses are claimed in the tax year the adoption is final. Domestic adoption expenses are allowed even when the adoption process is abandoned. Adoption expenses per child accumulate so any expenses taken for the credit in a prior year are evaluated in determining whether the maximum tax credit has been obtained. 
Qualified adoption expenses
Under US tax law, qualified expenses include adoption fees, court costs, attorney fees, traveling expenses (including amounts spent for meals and lodging while away from home), and other expenses directly related to and for which the principal purpose is the legal adoption of an eligible child.  The adoption tax credit is per child, thus the credit doubles when adopting two children in the same year. It is also important to note that this is a "credit," not a mere "deduction."  A tax credit is a dollar for dollar reduction of federal tax, not a reduction of taxable income, such as with a mortgage payment.
Parents who adopt a child with special needs can claim the full credit without documenting expenses. (See the IRS FAQs, paragraph 2 of question 1 at http://www.irs.gov/individuals/article/0,,id=231663,00.html.) There are several factors that determine if a child has "special needs" and those factors can vary by state.
A child with special needs has a factor or condition (uniquely defined by each State) that may involve any of the following:
Ethnic or racial background Age Membership in a sibling group Medical, physical, or emotional disabilities Risk of physical, mental, or emotional disability based on birth family history Any condition that makes it more difficult to find an adoptive family
These broader definitions of "special needs" may be used to determine eligibility for Federal financial assistance for adoption of children and youth from the U.S. foster care system. While there's no single Federal definition of special needs, according to title IV-E of the Social Security Act, a child or youth with special needs must also meet the following two requirements to be eligible for Federal adoption assistance:
1. The child or youth cannot or should not be returned home to his or her parent(s).
2. An unsuccessful attempt was made to place the child or youth without adoption (financial) assistance, except in cases where such a placement would not have been in the best interests of the child or youth.
Parents will need to document the child has special needs, and this documentation can include the adoption assistance/adoption subsidy agreement, a letter from the state/county approving the child for adoption assistance/adoption subsidy, or a letter from the state/county child welfare agency stating that the child has special needs. See question 13 at the FAQs for information about documentation.
For the 2012 tax filing season 90% of adoption tax credit claims were subject to IRS review and 69% were audited. The IRS Taxpayer Advocate Service cites this as a serious problem within the IRS. The average delay for these correspondence audits was 126 days. Over 55% of these correspondence audits were closed with no changes. 
state adoption tax benefits
Many states have adoption tax benefits that are in addition to the federal tax credit. These tax benefits are for state residents and claimed when they file there state tax returns:
California nonrefundable credit $2500 Georgia nonrefundable credit $2000 Kansas nonrefundable credit $1500 Mississippi nonrefundable credit $2500 Missouri nonrefundable credit $10000 Montana nonrefundable credit $1000 Ohio nonrefundable credit $1500 W Virginia nonrefundable credit $2000 Utah refundable credit $1000 N Carolina nonrefundable credit 50% of federal credit Arizona exclusion $3000 N Mexico exclusion $1000 N Dakota exclusion $1750 Oklahoma exclusion $20000 S Carolina exclusion $2000 Wisconsin exclusion $5000
Many of these states limit benefits to adoptions of children in foster care within the state. 
- "IRS Tax Topic 607".
- "IRS Adoption Tax Credit FAQ".
- "IRS 2012 form 8839 Instructions".
- Hicks, Randall. "ADOPTION: The Essential Guide to Adopting Quickly and Safely," Perigee Press 2007, pages 205-206.
- "US Dept. HHS Special Needs Adoption FAQ -- State List".
- "US Dept. HHS Special Needs Adoption FAQ".
- "iRS MSP 7 FAQ".
- "state adoption tax credit FAQ".