I'll describe a framework for studying what happens when one imposes various structural conditions on clustering schemes and tries to identify all methods that comply with such conditions. Within this framework, it is possible to prove a theorem analogous to one of J. Kleinberg, in which one obtains existence and uniqueness instead of a non-existence result. We also obtain a full classification of all clustering schemes satisfying a condition we refer to as excisiveness. The classification can be changed by varying the notion of maps of finite metric spaces and by doing so gives rise to families of clustering schemes that exhibit different degrees of sensitivity to density. Several of the constructions can carried out on non-symmetric spaces such as networks.