  This thesis contributes to the analytic theory of automorphic L-functions.
  We prove an approximate functional equation for the central value of the
L-series attached to an irreducible cuspidal automorphic representation of
GL(m) over a number field. We investigate the decay rate of the cutoff function
and its derivatives in terms of the analytic conductor introduced by Iwaniec
and Sarnak. We also see that the truncation can be made uniformly explicit at
the cost of an error term. The results extend to products of central values.
  We establish, via the Hardy-Littlewood circle method, a nontrivial bound on
shifted convolution sums of Fourier coefficients coming from classical
holomorphic or Maass cusp forms of arbitrary level and nebentypus. These sums
are analogous to the binary additive divisor sum which has been studied
extensively. We achieve polynomial uniformity in all the parameters of the cusp
forms by carefully estimating the Bessel functions that enter the analysis. As
an application we derive, extending work of Duke, Friedlander and Iwaniec, a
subconvex estimate on the critical line for L-functions associated to character
twists of these cusp forms.
  We also study the shifted convolution sums via the Sarnak-Selberg spectral
method. For holomorphic cusp forms this approach detects optimal cancellation
over any totally real number field. For Maass cusp forms the method is burdened
with complicated integral transforms. We succeed in inverting the simplest of
these transforms whose kernel is built up of Gauss hypergeometric functions.
