We define Ces{\`a}ro spaces on general measure spaces on which there is a pre-order provided. We find conditions under which these are quasi-Banach function spaces. Classical Ces{\`a}ro and Copson function and sequence spaces as well as some Morrey spaces appear as special cases of our general construction. We show that, under natural assumptions, the spaces are strictly convex, possess the diameter two property and contain asymptotically isometric copies of $\ell_1$ and do not contain isomorphic copies of $c_0$.