  This thesis is concerned with formulations of the Einstein equations in
axisymmetric spacetimes which are suitable for numerical evolutions. We develop
two evolution systems based on the (2+1)+1 formalism. The first is a
(partially) constrained scheme with elliptic gauge conditions arising from
maximal slicing and conformal flatness. The second is a strongly hyperbolic
first-order formulation obtained by combining the (2+1)+1 formalism with the Z4
formalism. A careful study of the behaviour of regular axisymmetric tensor
fields enables us to cast the equations in a form that is well-behaved on the
axis. Further topics include (non)uniqueness of solutions to the elliptic
equations arising in constrained schemes, and comparisons between various
boundary conditions used in numerical relativity. The numerical implementation
is applied to adaptive evolutions of nonlinear Brill waves, including twist.
