  The generalized Ces\`{a}ro operators $\mathcal{C}_t$, for $t\in[0,1)$,
introduced in the 1980's by Rhaly, are natural analogues of the classical
Ces\`{a}ro averaging operator $\mathcal{C}_1$ and act in various Banach
sequence spaces $X\subseteq {\mathbb C}^{{\mathbb N}_0}$. In this paper we
concentrate on a certain class of Banach lattices for the coordinate-wise
order, which includes all separable, rearrangement invariant sequence spaces,
various weighted $c_0$ and $\ell^p$ spaces
  and many others. In such Banach lattices $X$ the operators $\mathcal{C}_t$,
for $t\in[0,1)$, are always compact (unlike $\mathcal{C}_1$) and a full
description of their point, continuous and residual spectrum is given.
Estimates for the operator norm of $\mathcal{C}_t$ are also presented.
