Summary: The paper is concerned with \(L_p\)-estimates for solutions of the \(n\)-dimensional exterior Stokes problem. The main result of the paper are new \(L_p-L_q\) estimates \[ \begin{gathered} \bigl|v(t)\bigr |_q \leq C_1|v_0|_pt^{-\mu}, \tag{1}\\ \bigl|v_t(t) \bigr|_q\leq C_2 |v_0|_p t^{-\mu'}, \tag{2}\\ \bigl|\nabla v(t)\bigr|_q\leq C_3 |v_0|_p t^{-\widehat\mu}, \tag{3}\end{gathered} \] for the solution of a homogeneous Stokes problem with the initial condition \(v(x,0)=v_0(x)\); \(|\cdot|_p\) is the \(L_p\)-norm in an exterior domain \(\Omega\subseteq \mathbb{R}^n\). We prove that estimate (1) holds with \(\mu={n\over 2}({1\over p}-{1\over q})\) for arbitrary \(p,q\) satisfying the conditions \(1\leq p\leq q\leq\infty\), \(p+q>2\) for \(n>2\), \(10\). We show also that in formulas (1) and (3) \(\mu\), \(\widehat\mu\) are exact, in particular, that \(\widehat\mu n>2\). The method of the proof of (1)-(3) is quite elementary and relies on energy estimates, imbedding theorems, \(L_p-L_q\) estimates for the Cauchy problem and duality arguments. In addition, we give a new proof of \(W^{2,1}_{p,r} (Q_T)\) -- estimates of derivatives of the solution of the Stokes problem (here \(Q_T=\Omega \times(0,T)\); \(p,r>1)\), obtained by \textit{Y. Giga} and \textit{H. Sohr} [J. Funct. Anal. 102, 72-94 (1991; Zbl 0739.35067); and \textit{M. Giga, Y. Giga} and \textit{H. Sohr} [Functional Analysis and Related Topics, 1991, Lect. Notes Math. 1540, 55-67 (1993; Zbl 0804.47019)]. Inequality (1) allows us to show that the constant in this estimate can be taken independent of \(T\), if \(n>2\), \(p<{n\over 2}\), and we prove that the condition \(p< {n\over 2}\) can not be relaxed.