  This thesis examines edge-reinforced random walks with some modifications to
the standard definition. An overview of known results relating to the standard
model is given and the proof of recurrence for the standard linearly
edge-reinforced random walk on bounded degree graphs with small initial edge
weights is repeated. Then, the edge-reinforced random walk with multiple
walkers influencing each other is considered. The following new results are
shown: on a segment of three nodes, the edge weights resemble a P\'olya urn and
the fraction of the edge weights divided by the total weight forms a converging
martingale. On Z, the behavior is the same as for a single walker - either all
walkers have finite range or all walkers are recurrent. Finally,
edge-reinforced random walks with a bias in a certain direction are analysed,
in particular on Z. It is shown that the bias can introduce a phase transition
between recurrence and transience, depending on the strength of the bias, thus
fundamentally altering the behavior in comparison to the standard linearly
reinforced random walk.
