Infinite circuits in locally finite graphs With the naive definition of the cycle space most of the theorems concerning the cycle space become false in infinite graphs. This is exemplified by Tuttes generating theorem which states that the peripheral cycles generate the cycle space in a $3$ connected graph. To remedy this Diestel and Kühn proposed a topologically based definition of the cycle space in which the circles are precisely the homeomorphic images of the unit circle in the Freudenthal compactification of the (locally finite) graph. This notion not only includes the traditional finite cycles but also allows for certain infinite cycles. In the course of this thesis it is shown that the cycle space $\\mathcal C(G)$ of Diestel and Kühn is extremely fruitful and successful. Indeed it is demonstrated that the classical theorems about the cycle space carry over to locally finite graphs either in a verbatim manner or with only slight but obvious adaptions. The current work extends the following theorems to locally finite graphs: the planarity criteria of MacLane and Kelmans (the first of which solves a problem of Wagner 1970); Whitneys planarity criterium and duality in terms of spanning trees; and Gallais theorem. If infinite cycles are disallowed all of these results either fail completely or have to be weakened considerably. In finite graphs the elements of the cycle space are precisely those edge sets for which each vertex is incident with an even number of their edges. Such a characterisation based solely on vertex degrees becomes impossible in infinite graphs ...