The LOOK-COMPUTE-MOVE model for a set of autonomous robots has been thoroughly studied for over two decades. Each robot repeatedly LOOKS at its surroundings and obtains a snapshot containing the positions of all robots; based on this information, the robot COMPUTES a destination and then MOVES to it. Previous work assumed all robots are present at the beginning of the computation. What would be the effect of robots appearing asynchronously? This paper studies thisquestion, for problems of bringing the robots close together, andexposes an intimate connection with combinatorial topology. A central problem in the mobile robots area is the gathering problem. In its discrete version, the robots start at vertices in some graph G known to them, move towards the same vertex and stop. The paper shows that if robots are asynchronous and may crash, then gathering is impossible for any graph G with at least two vertices, even if robots can have unique IDs, remember the past, know the same names for the vertices of G and use an arbitrary number of lights to communicate witheach other. Next, the paper studies two weaker variants of gathering: edge gathering and 1-gathering. For both problems we present possibility and impossibility results. The solvability of edge gathering is fully characterized: it is solvable for three or more robots on a given graph if and only if the graph is acyclic. Finally, general robot tasks in a graph are considered. A combinatorial topology characterization for the solvable tasks is presented, by a reduction of the asynchronous fault-tolerant LOOK-COMPUTE-MOVE model to a wait-free read/write shared-memory computing model, bringing together two areas that have been independently studied for a long time into a common theoretical foundation.