  In this thesis, it is presented a set of results in adiabatic dynamics
(closed and open system) and transitionless quantum driving that promote some
advances in our understanding on quantum control and Hamiltonian inverse
engineering. In the context of adiabatic dynamics in closed systems, it is
introduced a validation mechanism for the adiabaticity conditions by studing
the system dynamics from a non-inertial reference frame. By considering a
decohering scenario, validity conditions of the adiabatic approximation are
also studied. As a fresh general result with potential applications, it is
shown that under decoherence the adiabaticity may still occur in the infinite
time limit, as it happens for closed systems, for a class of initial quantum
states. To end, the original contributions of this thesis to the theory of
shortcuts to adiabaticity refers to a generalized approach of transitionless
quantum driving, where one explores the gauge freedom of the quantal phase
factors accompanying adiabatic trajectories. A number of theoretical
applications are studied, where some theoretical prediction presented in this
thesis are experimentally verified through two different experimental setups,
namely a qubit encoded in the energy hyperfine structure of a Ytterbium trapped
ion, and in nuclear magnetic resonance with a nuclear spin qubit.
