The problem of predicting the future state of the weather and and climate of the earth has a long history in applied mathematics. Both problems were identified by John von Neumann as prime areas of study for the earliest electronic computers. Over the past 50 years, many mathematical challenges have been inspired by the problems associated with predicting weather and simulating the past present and future climate. Amongst these are stable integration of geophysical fluid equations for indefinite periods of time, the development of a theory of chaos and predictability and successful fusion of models and observational data in systems with extremely large dimensionality.

New challenges are arising in the 21st century which will require new computational and applied math techniques to resolve. I will present as survey of both the past successes and the future problems in my talk.