Abstract: The covering spectrum of a Riemannian manifold was first defined 
by Sormani-Wei in 2004. We will review the definition of this spectrum 
which roughly measures the sizes of holes in the space using a special 
sequence of covering spaces called delta covers. On compact spaces, we 
proved the covering spectrum is a subset of the (1/2) length spectrum and 
is determined by the marked length spectrum. This is not true on complete 
noncompact manifolds. In recent work with Wei, we have begun to prove 
related weaker theorems. To capture more information about complete 
manifolds we have defined two other spectra: the cut-off covering spectrum 
and the rescaled covering spectrum.