  In this dissertation we demonstrate that the continuous-time quantum walk
models remain powerful for nontrivial graph structures. We consider two aspects
of this problem. First, it is known that the standard Continuous-Time Quantum
Walk (CTQW), proposed by Childs and Goldstone, can propagate quickly on the
infinite path graph. However, the Schr\"odinger equation requires the
Hamiltonian to be symmetric, and thus only undirected graphs can be
implemented. In this thesis, we address the question, whether it is possible to
construct a continuous-time quantum walk on general directed graphs, preserving
its propagation properties. Secondly, the quantum spatial search defined
through CTQW has been proven to work well on various undirected graphs.
However, most of these graphs have very simple structures. The most advanced
results concerned the Erd\H{o}s-R\'enyi model of random graphs, which is the
most popular but not realistic random graph model, and Barab\'asi-Albert random
graphs, for which full quadratic speed-up was not confirmed. In the scope of
this aspect we analyze, whether quantum speed-up is observed for complicated
graph structures as well.
