The emerging sixth-generation (6G) wireless communication technologies such as terahertz communications, artificial intelligence (AI), and reconfigurable intelligent surfaces (RISs) require high data rates, low latency, and massive connectivity in real-life scenarios. In this context, wideband signal processing technologies have attracted increasing attention from both academia and industry. However, several challenges still exist, such as the need of initial estimates, a limited size of antenna/sensor array, unknown signal waveforms, frequency spectrum overlapping, a few snapshots, and grid mismatch problems. To tackle these problems, this thesis explores convex and nonconvex optimization methodologies for wideband signal direction of arrival (DOA) estimation. As a typical array signal processing technology for source signal localization and target tracking, DOA estimation has an extensive range of applications in radar, sonar, and wireless communications. The primary challenge in wideband signal DOA estimation lies in the varying steering matrices associated with different frequencies. To address this issue, this thesis proposes a gridless and covariance-free approach for jointly estimating the DOAs from multi-band signals with unknown waveforms. The greatest common divisor (GCD) of involved frequencies is employed to construct a unified frequency grid (UFG). Subsequently, a master steering matrix is formed, encompassing the steering matrices from different frequencies as submatrices. This innovative approach casts the wideband signal DOA estimation problem as a low-rank Hankel matrix recovery problem. Then, the corresponding convex and nonconvex optimization formulations are proposed, and algorithms are tailored to solve the specific problems. Numerical results showcase several advantages of the proposed approach, including the ability to estimate source angles exceeding the number of sensors in a uniform linear array (ULA), the elimination of the covariance matrix estimation step, robust performance even with only a few snapshots, fast convergence of nonconvex iterative solutions, and superior average root-mean-square error (RMSE) performance compared to the convex relaxation counterparts such as nuclear norm minimization (NNM).