As the simplest molecule in nature, the hydrogen molecular ion provides a unique system for studying molecules in intense laser fields. In this thesis, the Coulomb explosion and photodissociation channel of H2+ and D2+ have been investigated with 790-nm, sub-100-fs laser pulses by employing a high-resolution photofragment imaging technique. Unlike most experiments, where neutral molecules were used as a target, in this work molecular ions were prepared by an electric discharge, providing well-defined starting conditions and allowing experiments at lower intensities. At intensities close to the threshold for Coulomb explosion (10^14 W/cm^2), we observed a peak structure in the Coulomb explosion kinetic energy spectra in both H2+ and D2+. We show that the observed peaks can be attributed to the different dissociation energies of vibrationally excited molecules. Furthermore, this preservation of the vibrational structure during the Coulomb explosion suggests ionization at one well-defined critical internuclear distance. Moreover, when using pulses with durations of 200-500 fs, we found three Coulomb explosion kinetic energy groups with different angular distributions in both molecular ions. The groups in D2+ at a pulse duration of 350 fs and an intensity of 1x10^14 W/cm^2 strongly suggest critical internuclear distances of 8, 11 and 15 a.u. In the one-photon photodissociation study of D2+ at intensities below 2x10^14 W/cm^2, we obtained vibrationally resolved fragment velocity distributions. With increasing intensity, we observed the effects of bond softening - narrowing of the angular distributions and vibrational level shifting. These effects were also found in H2+ and are in agreement with the model of light-induced potentials. In addition, in the kinetic energy spectra of D2+, we found the smaller widths of the vibrational peaks, which we interpret as being due to the longer lifetimes of the vibrational states of D2+ against photodissociation. Moreover, we observed fragments with near-zero kinetic energies with broad angular distributions, indicating dissociation through a vibrational trapping mechanism. In the two-photon bond softening process (above-threshold dissociation) of H2+, we identified fragments from the single vibrational level v=3. At higher intensities, lower-energy fragments with increasing alignment were observed, suggesting dissociation of the levels v3 through the barrier lowering mechanism. These results provide a basis for quantum mechanical simulations that could lead to a better understanding of the molecular dynamics in intense fields.