The AlKr van der Waals complex was characterized by laser-induced fluorescence spectroscopy in a supersonic free jet. Rotationally resolved spectra were recorded for eight bands of the AlKr(B (2)SIGMA+ <-- X (2)PI(1/2)) transition and for seven bands of the AlKr(H (2)SIGMA+ <-- X (2)PI(1/2)) transition. Rydberg-Klein-Rees potential-energy curves were determined for the B (2)SIGMA+ and H (2)SIGMA+ states, using vibrational and rotational spectroscopic constants derived from the spectra. The interaction between the X (2)PI(1/2) ground state and the low-lying repulsive A (2)SIGMA+ state was characterized by analysis of the LAMBDA-doubling in the ground state. Equilibrium bond lengths were obtained for the X, B, and H states, and dissociation energies for the B and H states. The fact that the more strongly bound H (2)SIGMA+ state has a larger r(e) value than the B (2)SIGMA+ state was rationalized by postulating an avoided potential curve crossing between the H (2)SIGMA+ state and a repulsive valence (2)SIGMA+ state correlating with Al(3s(2)3d) + Kr.