For the Cesàro matrix C=(cnm)n,m∈N, where cnm=1n, if n≥ m and c_nm= 0 otherwise, the Cesàro sequence spaces ces0,cesp (for 1 < p< ∞) and ces_∞ are defined. These spaces turn out to be real vector lattices and with respect to a corresponding (naturally introduced) norm they are all Banach lattices, and so possess (or not possess) some interesting properties. In particular, the relations to their generating ideals c0,ℓp and ℓ_∞ are investigated. Finally the ideals of all finite, totally finite and selfmajorizing elements in ces, ces_p (for 1 < p< ∞) and ces_∞ are described in detail.