Ends of graphs Our topic is infinite graph theory with our focus on the ends of an infinite graph (which can be informally viewed as endpoints of rays) and their role in extensions of results known for finite graphs. Often these extensions fail if one does not take into account the ends of the graph but otherwise hold. In other cases results become more interesting when ends are considered as well as vertices. An example for the latter is the Erdös Menger conjecture for infinite graphs (recently proved by Aharoni and Berger): we shall prove a generalization which allows for ends in the considered paths and separators. This means that in an infinite graph we allow paths to be infinite. Moreover considering ends on a par with vertices we will allow these paths then called arcs to start or end in ends and to pass through them. Similarly the notion of a cycle will be generalized to that of a (possibly infinite) circle which may pass through ends. This leads to a different notion of forests (so called topological forests) in infinite graphs. Another aspect of the ends is that since in many ways they behave like vertices they should be attributed a degree. We introduce such a notion as well as a concept of parity for ends. For ends of finite degree the parity will coincide with the parity of the degree while ends of infinite degree will be classified into &lsquo;even&rsquo; and &lsquo;odd&rsquo;. Using these concepts (arcs circles topological forests degrees and parities of ends) we extend several results from finite graph theory verbatim to infinite graphs.