This paper reviews recent attempts to determine if some aspects of magnetospheric dynamics, and in particular substorms as measured by AE and AL geomagnetic indices, can be represented by a low-dimensional dynamical system. If true, this result would imply that a small set of ordinary differential equations could describe important aspects of substorm dynamics, greatly simplifying modeling and prediction efforts and providing significant input to more detailed modeling. The ''embedding'' and ''correlation dimension'' methods used to investigate the dimensionality of a physical process from a single time series are considered in detail with an emphasis on what can go wrong and what can be done about it. Two main conclusions of this work, which includes some new results on the particular case of AL, are (1) that a low-dimensional and probably strange attractor does exist in magnetospheric dynamics, and (2) that there is no reliable substitute for using large numbers of data points in performing analyses leading to this conclusion. ¿American Geophysical Union 1991