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\centerline{FROM METRIC SPACES TO EINSTEIN SPACES}    
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Gromov defined a notion of distance between isometry classes of compact 
metric spaces.  Thus, the collection of all such isometry classes is itself, 
a metric space, ${\cal M}$.  Roughly speaking, 
two metric spaces are close in ${\cal M}$ if, ``to the
naked eye'', they look the same (although when examined on a 
microscopic scale, they might look entirely different). 

Gromov showed that a certain important class 
of smoothly curved spaces, ${\cal RIC}_n\subset{\cal M}$, consisting of 
$n$-dimensional riemannian manifolds with a definite bound on diameter 
and lower bound on Ricci curvature, has the property that every sequence, 
$\{M^n_i\}$, with $M^n_i\in{\cal RIC}_n$, has a subsequence which converges 
to some metric space $Y\in \overline{{\cal RIC}_n}\subset {\cal M}$. 
Because of the weak nature of the convergence, apriori,
$Y$ might not resemble a smoothly curved space in any reasonable way. 
On the other hand, knowledge of the extent to which 
$Y\in\overline{{\cal RIC}_n}$
could fail to be smoothly curved, would provide information
on the class, ${\cal RIC}_n$, itself. How can one get such information?  

Starting with simple examples, we will discuss curvature and  Gromov's
notion of convergence of metric spaces.  Then we will describe 
a theory (developed jointly with T. Colding) which governs the regularity and 
singularity structure of spaces $Y\in\overline{{\cal RIC}_n}$.  
If we specialize further to the case of Einstein manifolds, spaces whose
Ricci tensor is a constant multiple of the metric, stronger conclusions
can be obtained.  In this way, we get information on the manner in which 
Einstein metrics can degenerate. 
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\noindent
{\bf Lecture 1:} Curvature and Gromov-Hausdorff convergence.  
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{\bf Lecture 2:} Spaces with Ricci curvature bounded below. 
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{\bf Lecture 3:} Degeneration of Einstein metrics. 



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