a node are locally labeled by distinct positive integers. Degrees of nodes can be either finite or infinite. No co- herence between those local labelings is assumed. When an agent leaves a node, it is aware of the port number by which it leaves and when it enters a node, it is aware of the entry port number. It can also verify, at each node, whether a given positive integer is a port num- ber at this node. Agents know neither the graph, nor the initial distance between them. They cannot mark the nodes or the edges in any way. Rendezvous has to be accomplished regardless of local labelings of ports. Each agent terminates its walk at the time of meeting the other agent. In the geometric scenario, we assume that the terrain in which agents operate is a closed subset of the Euclidean plane, i.e., it contains limits of all converging sequences of points in it. The boundary of the terrain is defined as the set of points having arbitrarily close points both in the terrain and outside of it. Since the terrain is closed, the boundary is included in it. All other points of the terrain are its interior points. Each agent can only distinguish if it is currently in an interior point of the terrain or in its boundary. Agents do not know the terrain in which they operate, they cannot “see” any vicinity of the currently visited point and they cannot leave any marks. Agents are equipped with a compass and with the same unit of length. Thus systems of coordinates of agents are aligned and the origin for each agent is at its starting point. Again, an agent terminates its walk at the time of meeting the other agent. If agents are identical, i.e., they do not have dis- tinct identifiers, and execute the same algorithm, then deterministic rendezvous is impossible, e.g., in the ring (graph scenario) or in the plane (geometric scenario): the adversary will make the agents move always in the same direction at the same speed, keeping them at the same distance at all times, thus they will never meet. Hence we assume that agents have distinct identifiers, called labels, which are two different positive integers. This is their only way to break symmetry. We assume that each agent knows its own label but not the la- bel of the other agent. This excludes, e.g., rendezvous strategies of the type “waiting for mommy”, in which the agent with smaller label remains idle and the other agent explores the graph or the terrain. We do not im- pose any restriction on the memory of the agents: from the computational perspective they are viewed as Tur- ing machines. Two important notions used to describe movements of agents are the route of each agent and its walk . Roughly speaking, each agent chooses the route where it moves and the adversary describes the walk on this route, deciding how the agent moves. More precisely, these notions are defined as follows. The adversary initially places an agent with label ℓ at some node of the graph or at some point in the terrain. Given this label and this starting point, the route is chosen by the agent and is defined as follows. In the case of the graph, the agent chooses one of the available ports at the current node. After getting to the other end of the corresponding edge, the agent chooses one of the available ports at this node, and so on, indefinitely (until rendezvous). The resulting route of the agent is the corresponding sequence of edges, which is a (not necessarily simple) path in the graph. The route in a terrain is a sequence ( S 1 , S 2 , . . . ) of segments, where S$_{i}$ = [ a$_{i}$, a$_{i}$ +1 ], defined in stages as follows, given the agent’s label and the starting point. In stage i the agent starts at point a$_{i}$ , and a 1 is the starting point chosen by the adversary. The agent chooses a direction α and distance x . If the segment of length x in direction α starting in v intersects the boundary of the terrain at some distance y ≤ x from v , the agent becomes aware of it in the intersection point w closest to v . In this case, the stage ends at w and in the next stage the agent chooses the reverse direction α and the distance y (that will cause it to return to v ). If the segment of length x in direction α starting in v does not intersect the boundary of the terrain, the stage ends when the agent reaches point u at distance x from v in direction α . Stages are repeated indefinitely (until rendezvous). We now describe the walk f of an agent on its route. Let R = ( S 1 , S 2 , . . . ) be the route of an agent. In the graph scenario this is a (not necessarily simple) infinite path in the (spatial embedding of) the graph, and in the geometric scenario it is an infinite polygonal line in the plane. Let ( t 1 , t 2 , . . . ), where t 1 = 0, be an increasing sequence of reals, chosen by the adversary, that represent points in time. Let f$_{i}$ : [ t$_{i}$, t$_{i}$ +1 ] → [ a$_{i}$, a$_{i}$ +1 ] be any continuous function, chosen by the adversary, such that f$_{i}$ ( t$_{i}$ ) = a$_{i}$ and f$_{i}$ ( t$_{i}$ +1 ) = a$_{i}$ +1 . For any t ∈ [ t$_{i}$, t$_{i}$ +1 ], we define f ( t ) = f$_{i}$ ( t ). The interpretation of the walk f is as follows: at time t the agent is at the point f ( t ) of its route. This general definition of the walk and the fact that it is constructed by the adversary capture the asynchronous characteristics of the process. The movement of the agent can be at arbitrary speed, the agent may sometimes stop or go back and forth, as long as the walk in each segment of the route is continuous and covers all of it. Notice that the power of the asynchronous adver- sary to produce any continuous walk on the routes de- termined by the agents implies the following signifi- cant difference with respect to the synchronous scenario.