  Two mobile agents (robots) with distinct labels have to meet in an arbitrary,
possibly infinite, unknown connected graph or in an unknown connected terrain
in the plane. Agents are modeled as points, and the route of each of them only
depends on its label and on the unknown environment. The actual walk of each
agent also depends on an asynchronous adversary that may arbitrarily vary the
speed of the agent, stop it, or even move it back and forth, as long as the
walk of the agent in each segment of its route is continuous, does not leave it
and covers all of it. Meeting in a graph means that both agents must be at the
same time in some node or in some point inside an edge of the graph, while
meeting in a terrain means that both agents must be at the same time in some
point of the terrain. Does there exist a deterministic algorithm that allows
any two agents to meet in any unknown environment in spite of this very
powerfull adversary? We give deterministic rendezvous algorithms for agents
starting at arbitrary nodes of any anonymous connected graph (finite or
infinite) and for agents starting at any interior points with rational
coordinates in any closed region of the plane with path-connected interior.
While our algorithms work in a very general setting ? agents can, indeed, meet
almost everywhere ? we show that none of the above few limitations imposed on
the environment can be removed. On the other hand, our algorithm also
guarantees the following approximate rendezvous for agents starting at
arbitrary interior points of a terrain as above: agents will eventually get at
an arbitrarily small positive distance from each other.
