This thesis discusses various aspects of nodal sets of random Gaussian Laplace eigenfunctions (‘arithmetic random waves’) on the two- and three-dimensional tori. The first problem concerns the number of nodal intersections against a straight line segment in two dimensions. The expected intersections number, against any smooth curve, is universally proportional to the length of the ref-erence curve, times the wavenumber, independent of the geometry. I bounded the variance in the case of a straight line with rational slope. Without assuming rational slope, I proved that the same bound holds unconditionally for a density one sequence of energies, and conditionally for all energies. The three-dimensional analogue of the first problem is the study of the nodal intersections variance against a straight line segment on the three dimensional torus. I gave a bound for rational lines. For irrational lines, I proved an uncon-ditional result, and a stronger conditional result. I also found a better bound for irrational lines (a1, a2, a3) where a2/a1 is rational. The third problem is work in collaboration with J. Benatar. We studied the area of the nodal set in the three dimensional case. The expected area is proportional to the square root of the eigenvalue. We established an asymptotic formula for the nodal area variance. The methods involve the theory of random processes, the study of the covari-ance function and application of Kac-Rice formulas. The problems are closely related to the theory of lattice points on circles and spheres. I proved upper bounds for the number of lattice points on spheres that lie on a thin spher-ical segment, using Diophantine approximation. Together with J. Benatar, I bounded the number of non-degenerate 4-correlations, and 6-correlations, of lat-tice points on spheres.