Photoprocesses are ubiquitous in nature, science, and engineering. The understanding as well as the optimization of photochemical and photophysical properties of molecular systems requires computational tools that are able to describe the dynamical evolution of the system in electronically excited states. Ab Initio Molecular Dynamics (AIMD) based on Density Functional Theory (DFT) has become an established tool for elucidating mechanisms of chemical reactions that occur in the electronic ground state. However, to describe photoprocesses by AIMD, an underlying electronic-structure method that is able to treat excited states is necessary. This complicates the description of these processes because in the past this implied the use of computationally expensive wavefunction-based methods, which in addition are not straightforward to use. Time-Dependent Density Functional Theory (TDDFT) provides an in principle exact description of electronically excited states, although in practice, approximations have to be introduced. Compared to wavefunction-based methods, TDDFT is computationally less demanding and is relatively straightforward and easy to use. Recently, TDDFT nuclear gradients have become available and allow to carry out AIMD in excited states. In this thesis a TDDFT-based AIMD method that is able to account for non-adiabatic effects is developed and implemented. The non-adiabatic couplings are computed by means of a Kohn-Sham orbital based reconstruction of the many-electron wavefunction for ground and excited states. The non-adiabatic scheme is based on the fewest switches trajectory surface hopping (SH) method introduced by Tully. The method is applied to describe decay processes, such as fragmentation or isomerization, that occur upon photoexcitation of the molecules protonated formaldimine and oxirane. In the case of protonated formaldimine, the results of the TDDFT-SH method are in good agreement with SH simulations based on the state-averaged complete active space (SA-CASSCF) method, both with respect to the observed reaction mechanisms and the excited state life times. In the case of oxirane, the TDDFT-SH simulations confirm the main experimental results and provide an additional refinement of the postulated reaction mechanism. The accuracy of TDDFT is investigated with respect to different issues that are especially important for the proper description of photoprocesses. These aspects include the accuracy of non-adiabatic coupling (NAC) vectors, the description of S1-S0 conical intersections, and the description of locally excited states in systems where charge transfer (CT) states are present, that are affected by the well-known CT failure of TDDFT. Concerning the NAC vectors, a qualitative agreement with SA-CASSCF is found, although magnitudes are underestimated by TDDFT/PBE. Regarding the description of conical intersections to the ground state, we find as expected that TDDFT in the adiabatic approximation (ALDA) is not able to predict an intersection that is strictly conical. However, we find that TDDFT is able to approximate a conical intersection that has a similar shape as the one predicted by SA-CASSCF. For an electron donor-bridge-acceptor molecule it is shown that the CT failure of TDDFT can also considerably affect properties of non-CT states. The use of TDDFT using conventional exchange-correlation functionals is thus not recommended for the description of such systems. Using the second-order approximate coupled cluster (CC2) method in conjunction with a high quality basis set, an accurate and balanced description of both locally excited and CT states can be made. The use of CC2 with large basis sets for AIMD simulations is however still computationally unaffordable for larger systems. TDDFT is still in its infancy and several attempts to cure some of its defiencies have already been made. These attempts mainly concern improvements of the approximations of the exchange-correlation functionals and associated TDDFT kernels. The TDDFT-SH method that has been developed in this thesis can in principle be applied in combination with any approximation for the exchange correlation functional, provided that nuclear gradients for this approximation are available and the computational cost remains acceptable. In this way, the method developed here is able to directly profit from the ongoing improvements in the active research field of exchange-correlation functionals and kernels.