A fractal structure over a topological space \(X\) is a family of closed coverings \(\boldsymbol\Gamma=\{\Gamma_n:n\in\omega\}\) such that \(\Gamma_{n+1}\) refines \(\Gamma_n\), the family \(\mathcal U_x\) of sets \(U_{xn}=\bigcup\{A\in G_n:x\in A\}\setminus \bigcup\{A\in G_n:x\notin A\}\) for \(n\in\mathbb N\) is an open neighbourhood base of \(x\) for every \(x\in X\), and \(x\in A\in\Gamma_n\) implies that there is \(B\in\Gamma_{n+1}\) such that \(x\in B\subseteq A\). The couple \((X,\boldsymbol\Gamma)\) is said to be a generalized fractal space, shortly \mathit{GF}-space. The authors study various notions of completeness in GF-spaces including \(K\)-completeness, bicompleteness, convergence completeness, half-completeness, right \(K\)-sequentially completeness, and other defined by themselves.