We have examined the evolution of the concept of a Hebbian synaptic modification and have suggested a contemporary definition. The biophysical mechanism demonstrated in vitro to control the induction of one type of hippocampal LTP has been shown to satisfy our definition of a Hebbian synaptic modification. Whether this biophysical mechanism is involved in the organization of behavior in the manner that Hebb originally envisioned remains to be seen. We have also summarized several modification algorithms that have been explored in theoretical studies of learning in adaptive networks. These algorithms also satisfied our definition of a Hebbian modification, but their relationships to known neurobiology require further exploration. By reviewing the biophysical mechanisms and formal algorithms together, we have exposed obvious similarities and differences. Such comparisons may help bridge the gap between computational theory and knowledge of the neurobiology of use-dependent synaptic change. Current models of LTP reveal that the activity-modification relationships are extremely sensitive to the biophysical/molecular details. The activity-modification relationships obviously can have a major influence on adaptive neurodynamics at the network level. As more accurate representations of the biological complexity and diversity are introduced into adaptive network simulations, we expect to gain new insights into the classes of computation that particular networks are capable of performing.