Gaussian processes are the basis of much statistical modeling  for spatial and spatio-temporal data.  The covariance structure
of the model generally needs to be estimated from the available  data and likelihood-based methods are a natural choice.  When the
 data do not fall on a regular grid, exact calculation of  the likelihood function typically requires computations on the
 order of the cube of the sample size and memory on the order of the square of the sample size.  In present computing environments,
 it is often the memory requirement that is the main bottleneck. This talk summarizes various approaches to reducing computations
 and memory and describes some recent efforts by myself, Jie Chen and Mihai Anitescu to calculate maximum likelihood estimates
 without ever needing the log determinant of the covariance matrix that appears in the loglikelihood function.  The algorithm requires
 only solutions of linear systems and the memory requirements are linear in the number of observations.  I will briefly describe
 a possible approach to approximating likelihoods for datasets that are too large even to allow algorithms 
that are linear in the number of observations.