I have moved for private reasons. Can I claim the moving costs against tax?

Privately induced moving costs are eligible as household-related services according to § 35a EStG. The prerequisite for the tax reduction according to § 35a EStG is that you can prove the expenses by presenting an invoice and the payment to the account of the service provider by a bank receipt.

Business/professional relocation costs continue to be deductible as business expenses or income-related expenses (possibly limited to a maximum amount).