Let \(\Lambda\) be a finite dimensional hereditary k-algebra and TrD the inverse of the Auslander-Reiten translation DTr. In the paper, the preprojective algebra \(\Pi\) (\(\Lambda)\) of \(\Lambda\) is the Z-graded algebra \(\Pi (\Lambda)=\oplus^{\infty}_{n=0}\Pi_ n(\Lambda)\), where \(\Pi_ n(\Lambda)=Hom_{\Lambda}(\Lambda,(TrD)^ n(\Lambda))\) and multiplication is given by \(u_ r\cdot u_ s=(TrD)^ r(u_ r)\circ u_ s\). It is shown that, if \(\Lambda\) is tame, then \(\Pi\) (\(\Lambda)\) is a prime finitely generated k-algebra, satisfies a polynomial identity, is left and right Noetherian of Krull dimension and global dimension two, and has zero Jacobson radical. The homogeneous prime spectrum of \(\Pi\) (\(\Lambda)\) is also described. In the proofs, the functorial methods are essentially applied. If \(\Lambda\) is wild, the homological properties of \(\Pi\) (\(\Lambda)\) are described by D. Baer [Lect. Notes Math. 1177, 1-12 (1986)].