Can 2 people claim a child on their tax return?

In general, the answer is no. The IRS only allows one person to claim a child with a unique social security number on their taxes, and get the credits that go with the exemption. However, there are exceptions for children of divorced or separated parents. Only one parent may claim the social security number, but the custodial parent may use the child in order to claim the head of household status, to receive the earned income credit and to receive the credit for child and dependent care expenses - even if they did not claim the exemption for the child.