  Manifolds with boundary and with corners form categories ${\bf
Man}\subset{\bf Man^b}\subset{\bf Man^c}$. A manifold with corners $X$ has two
notions of tangent bundle: the tangent bundle $TX$, and the b-tangent bundle
${}^bTX$. The usual definition of smooth structure uses $TX$, as
$f:X\to\mathbb{R}$ is defined to be smooth if $\nabla^kf$ exists as a
continuous section of $\bigotimes^kT^*X$ for all $k\ge 0$.
  We define 'manifolds with analytic corners', or 'manifolds with a-corners',
with a different smooth structure, in which roughly $f:X\to\mathbb{R}$ is
smooth if ${}^b\nabla^kf$ exists as a continuous section of
$\bigotimes^k({}^bT^*X)$ for all $k\ge 0$. These are different from manifolds
with corners even when $X=[0,\infty)$, for instance
$x^\alpha:[0,\infty)\to\mathbb{R}$ is smooth for all real $\alpha\ge 0$ when
$[0,\infty)$ has a-corners. Manifolds with a-boundary and with a-corners form
categories ${\bf Man}\subset{\bf Man^{ab}}\subset{\bf Man^{ac}}$, with well
behaved differential geometry.
  Partial differential equations on manifolds with boundary may have boundary
conditions of two kinds: (i) 'at finite distance', e.g. Dirichlet or Neumann
boundary conditions, or (ii) 'at infinity', prescribing the asymptotic
behaviour of the solution. We argue that manifolds with corners should be used
for (i), and with a-corners for (ii). We discuss many applications of manifolds
with a-corners in boundary problems of type (ii), and to singular p.d.e.
problems involving 'bubbling', 'neck-stretching' and 'gluing'.
