This paper is devoted to the study of tractability of the $L_2$-approximation and integration from weighted Korobov spaces of increasing smoothness in the worst-case setting. The considered algorithms use information from the class ${\mathrm{\Lambda }}^{\mathrm{a}\mathrm{l}\mathrm{l}}$, including all continuous linear functionals, and from the class ${\mathrm{\Lambda }}^{\mathrm{s}\mathrm{t}\mathrm{d}}$, including function evaluations. Necessary and sufficient conditions on the weights of the function space for strong polynomial tractability, polynomial tractability, quasi-polynomial tractability, uniform weak tractability, weak tractability and $(\sigma ,\tau )$-weak tractability, are provided. Our results give a comprehensive picture of the weight conditions for all standard notions of algebraic tractability. It may be helpful to study the tractability of nonhomogeneous tensor product spaces.