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We investigate the isomorphic structure of the Ces\`aro spaces and theirduals, the Tandori spaces. The main result states that the Ces\`aro functionspace $Ces_{\infty}$ and its sequence counterpart $ces_{\infty}$ areisomorphic, which answers to the question posted in \cite{AM09}. This is rathersurprising since $Ces_{\infty}$ has no natural lattice predual similarly as theknown Talagrand's example \cite{Ta81}. We prove that neither $ces_{\infty}$ isisomorphic to $l_{\infty}$ nor $Ces_{\infty}$ is isomorphic to the Tandorispace $\widetilde{L_1}$ with the norm $\|f\|_{\widetilde{L_1}}=\|\widetilde{f}\|_{L_1},$ where $\widetilde{f}(t):= \esssup_{s \geq t} |f(s)|.$Our investigation involves also an examination of the Schur and Dunford-Pettisproperties of Ces\`aro and Tandori spaces. In particular, using Bourgain'sresults we show that a wide class of Ces{\`a}ro-Marcinkiewicz andCes{\`a}ro-Lorentz spaces have the latter property.

Isomorphic structure of Ces\`aro and Tandori spaces