O-minimal structures on the real field have many desirable properties. As examples:

(1) For each positive integer p, every closed definable set is the zero set of a definable total C^p function.

(2) All compact definable C^0 submanifolds have finite volume.

(3) All metric dimensions commonly enountered in geometric measure theory, fractal geometry and analysis on metric spaces agree with topological dimension on coordinate projections of closed definable sets. (This can be made precise.)

(4) Connected components of coordinate projections of closed definable sets are definable.

(5) Closed definable sets have few (in the sense of Pila and Wilkie) rational points.

But o-minimality is not necessary for any of the above to hold. I will illustrate via examples as to why we might care about the analytic geometry of structures on the real field that are not necessarily o-minimal.