When a sufficiently strong laser field acts on an atom or a molecule, ionization can occur. Electrons released in this process are accelerated by the laser field and the distribution of their final momenta can be measured. As opposed to using linear or circular polarization to drive the ionization process, tailored fields provide additional degrees of freedom to create field shapes with special properties. The present thesis investigates the interaction of atoms and molecules with such fields through numerical calculation of photoelectron momentum distributions, and their application towards a time-resolved picture of strong-field ionization.For a two-color scheme where a weak orthogonal second harmonic is used to probe the ionization process in a strong linearly polarized laser field by observing the modulation of the signal as a function of the delay between the two colors, we solve the time-dependent Schrödinger equation in three dimensions and find the time of ionization resolved by final photoelectron momentum. We demonstrate that the delay scan is sensitive to Coulomb focusing and reveals signatures of photoelectron holography.While two-color schemes can be used to measure ionization times in linear polarization, the attoclock is used in circular polarization. There, the ionization time is inferred from the detection angle of the photoelectron. Because of Coulomb effects, a theoretical model is always required to determine the precise mapping. Contrary to models that are typically used, we obtain this mapping without relying on the notion of an electron trajectory. This is achieved by considering the stationary points of the Dyson integral representation of the time-dependent Schrödinger equation. We find these stationary points using numerical wave function propagation in complex time and confirm that the maximum of the momentum distributions corresponds well to the time of peak field strength.Using a counter-rotating bicircular laser field, the concept of the attoclock can be transferred to other types of polarization. For suitable field strength ratio, the electric field is approximately linearly polarized around the time of peak field strength while the shape of the vector potential is similar to the attoclock. First, we apply the trajectory-free theory to this field to find the most probable time of ionization. Second, we combine the bicircular field with the two-color scheme. This allows us to compare the ionization time measured in the two-color scheme with the one measured in the attoclock. We find that the orthogonal two-color scheme measures ionization time as if the Coulomb potential were not present. However, switching to parallel polarization, we obtain meaningful ionization times in accordance with the attoclock principle that ionization takes place most likely at the peak of the pulse. Applying the bicircular field to a polar molecule, we find that the momentum distribution shows a dependence on the orientation, but this does not imply an orientation dependence of the ionization time.