Treasure hunt and rendezvous are fundamental tasks performed by mobile agents in graphs. In treasure hunt, an agent has to find an inert target (called treasure) situated at an unknown node of the graph. In rendezvous, two agents, initially located at distinct nodes of the graph, traverse its edges in synchronous rounds and have to meet at some node. We assume that the graph is connected (otherwise none of these tasks is feasible) and consider deterministic treasure hunt and rendezvous algorithms. The time of a treasure hunt algorithm is the worst-case number of edge traversals performed by the agent until the treasure is found. The time of a rendezvous algorithm is the worst-case number of rounds since the wakeup of the earlier agent until the meeting. To the best of our knowledge, all known treasure hunt and rendezvous algorithms rely on the assumption that degrees of all nodes are finite, even when the graph itself may be infinite. In the present paper we remove this assumption for the first time, and consider both above tasks in arbitrary connected graphs whose nodes can have either finite or countably infinite degrees. Our main result is the first universal treasure hunt algorithm working for arbitrary connected graphs. We prove that the time of this algorithm has optimal order of magnitude among all possible treasure hunt algorithms working for arbitrary connected graphs. As a consequence of this result we obtain the first universal rendezvous algorithm working for arbitrary connected graphs. The time of this algorithm is polynomial in a lower bound holding in many graphs, in particular in the tree all of whose degrees are infinite.