It is impossible to correctly understand cometary activity without a fully reliable model of the nucleus surface activity. Thermal models of the nucleus interior, and gasdynamic models of the coma need to incorporate a surface activity model. Phenomenological models of the coma structures claim to deduce surface properties. In view of the presently developed future comet nucleus and landing missions, this field is expected to become central in cometary science.

A brief review of the recent cometary literature revels that, in nearly all works in which the surface activity is advocated explicitly or implicity, irrealistic over-simplified (and often inconsistent) assumptions are made. In the few works where an attempt is made to use carefully justified physical principles, the full arsenal of existing mathematical methods is far from having been used, so that the image of the surface activity is far behind what it could be. It means that this field of investigation is nearly fully open. Here, we try to list the problems encountered, and to qualify their present state of understanding. We address (1) to the microphysics of the gas and dust production and their modelling; (2) to the problem of allowing for the surface orography; (3) to the secular evolution of the surface composition and shape under the effect of its own activity; (4) to the coupling between surface activity and nucleus motion-around-center-of-mass (``rotation''). We point out the urgent need for a coupling between nucleus interior modeling and near-nucleus coma modeling, and, in first place, the need for a common representation of the properties of the surface.

The problem of matching surface activity models with present ground-based and future near-nucleus observations is discussed: in particular, we discuss (1) the relation between circumnuclear gas and dust distribution, and nucleus surface properties; (2) the urgent need for a matching between surface activity model results and observed nuclei non-gravitational force and ``rotation'' state; (3) the problem of deriving simple, but non-misleading parameters for statistical representations of the nuclei in general, from routine ground-based observations of their activity.