We consider the Volterra integral operator \(T_{\rho,\psi}:L_p(0,\infty)\to L_q(0,\infty)\) for \(1\leq p,q\leq \infty\), defined by \((T_{\rho,\psi}f)(s) =\rho(s)\int_0^s \psi(t) f(t) dt\) and investigate its degree of compactness in terms of properties of the kernel functions \(\rho\) and \(\psi\). In particular, under certain optimal integrability conditions the entropy numbers \(e_n(T_{\rho,\psi})\) satisfy \( c_1\Vert{\rho\,\psi}\Vert_r\leq \liminf_{n\to\infty} n\, e_n(T_{\rho,\psi}) \leq \limsup_{n\to\infty} n\, e_n(T_{\rho,\psi})\leq c_2\Vert{\rho\,\psi}\Vert_r\) where \(1/r = 1- 1/p +1/q >0\). We also obtain similar sharp estimates for the approximation numbers of \(T_{\rho,\psi}\), thus extending former results due to Edmunds et al. and Evans et al.. The entropy estimates are applied to investigate the small ball behaviour of weighted Wiener processes \(\rho\, W\) in the \(L_q(0,\infty)\)–norm, \(1\leq q\leq \infty\). For example, if \(\rho\) satisfies some weak monotonicity conditions at zero and infinity, then \(\lim_{\varepsilon\to 0}\,\varepsilon^2\,\log\mathbb{P}(\Vert{\rho\, W}\Vert_q\leq \varepsilon) = -k_q\cdot\Vert{\rho}\Vert_{{2q}/{2+q}}^2\).