We study quantum dynamical properties of discrete one-dimensional models of Schrodinger operators. The first goal of this thesis is to understand the time evolution of initial states supported on more than one site. We develop tools to bound the so-called transport exponents both from below and from above and apply them to several models. In particular we extend a group of results concerning Sturmain Hamiltonians, quasi-periodic Hamiltonians, substitution generated models and random polymer model. The second topic is a detailed analysis of the transport exponents in the case of a Sturmain model when the frequency is a quadratic irrational. In this case methods from hyperbolic dynamics are applied to study the trace map of the operator. We show that the wavepacket spreads out with the same polynomial rate on all possible timescales. The last model of interest is the Anderson model. We provide a new proof of the Anderson localization phenomenon and derive dynamical localization bounds for states supported on more than one site. This thesis contains joint work with Valmir Bucaj, David Damanik, Jake Fillman, Tom VandenBoom, Fengpeng Wang, and Zhenghe Zhang.