In this thesis the properties of two kinds of non-uniform random recursive trees are studied. In the first model weights are assigned to each node, thus altering the attachment probabilities. We will call these trees weighted recursive trees. In the second model a different distribution rather than the uniform one is chosen on the symmetric group, namely a riffle shuffle distribution. These trees will be called biased recursive trees. For both of these models the number of branches, the number of leaves, the depth of nodes and some other properties are studied. The focus is on asymptotic results and the comparison with uniform random recursive trees. It will be shown that the studied properties of weighted recursive trees are close to uniform recursive trees in many cases when the number of nodes increases. In contrast biased recursive trees show a different behaviour but approach uniform recursive trees depending on the parameters of the riffle shuffle distribution.