Let \(\mathcal C\) be the abelian category of complexes of left \(R\)-modules. The complex \(F \in \mathcal C\) is said to be flat if it is a direct limit of finitely generated projective complexes. Some characterizations and properties of flat complexes are investigated in section 2. If \(\mathcal A \subseteq \mathcal C\) is a class of complexes, then a morphism \(\varphi:E\to X\), \(E \in \mathcal A\), is called an \(\mathcal A\)-precover of \(X\), if for each morphism \(\varphi' :E'\to X\), \(E' \in \mathcal A\), there exists a morphism \(\psi : E' \to E\) such that \(\varphi \psi = \varphi'\). Morever, \(\varphi\) is an \(\mathcal A\)-cover of \(X\) whenever every endomorphism \(\psi\) of \(E\) with \(\varphi\psi = \varphi\) is an automorphism of \(E\). Basic properties of flat precovers are presented in the part 3. Recall that a complex \(F\) is called \(DG\)-flat if all its members \(F^n\) are flat \(R\)-modules. In the last two sections the existence of flat and DG-flat covers of complexes over a commutative noetherian ring of finite Krull dimension is proved (theorem 4.6 and theorem 5.4.).