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We discuss the existence of the global attractor for a family of processes$U_\sigma(t,\tau)$ acting on a metric space $X$ and depending on a symbol$\sigma$ belonging to some other metric space $\Sigma$. Such an attractor isuniform with respect to $\sigma\in\Sigma$, as well as with respect to thechoice of the initial time $\tau\in\R$. The existence of the attractor isestablished for totally dissipative processes without any continuityassumption. When the process satisfies some additional (but rather mild)continuity-like hypotheses, a characterization of the attractor is given.

Totally dissipative dynamical processes and their uniform global attractors