The objective of this thesis is to develop gridless super-resolution Direction-of-Arrival (DOA) estimation methods for arbitrary array geometries which exploit sparsity in the continuous parameter space to eliminate the offgrid problem associated with grid-based compressed sensing methods. The focus is on designing search-free gridless DOA methods that are effective in terms of achieving higher resolution and accuracy. We also focus on designing efficient methods, requiring fewer sensors and bits in data representation to reduce the hardware complexity of sensor arrays. Making use of the periodicity of the array manifold, the dual function for the infinite-dimensional primal atomic norm minimization problem is represented as a trigonometric polynomial via truncated Fourier series. The dual problem is then converted to a finite semidefinite program, and the source directions are recovered through polynomial rooting. The proposed approach is used to design search-free gridless methods applicable for coherent sources, limited snapshots, and one-bit sensor measurements. Extensions to improve the degrees of freedom using spatial correlations and higher order statistics are also developed. The improved performance of the proposed method is demonstrated by computer simulations for various array geometries and parameters.