In this paper, we systematically investigate the feasibility of different extremum-seeking (ES) control and optimization schemes to improve the conversion efficiency of wave energy converters (WECs). Continuous-time and model-free ES schemes based on the sliding mode, relay, least-squares gradient, self-driving, and perturbation-based methods are used to improve the mean extracted power of a heaving point absorber subject to regular and irregular waves. This objective is achieved by optimizing the resistive and reactive coefficients of the power take-off (PTO) mechanism using the ES approach. The optimization results are verified against analytical solutions and the extremum of reference-to-output maps. The numerical results demonstrate that except for the self-driving ES algorithm, the other four ES schemes reliably converge for the two-parameter optimization problem, whereas the former is more suitable for optimizing a single parameter. The results also show that for an irregular sea state, the sliding mode and perturbation-based ES schemes have better convergence to the optimum in comparison to other ES schemes considered here. The convergence of PTO coefficients toward the performance-optimal values is tested for widely different initial values in order to avoid bias toward the extremum. We also demonstrate the adaptive capability of ES control by considering a case in which the ES controller adapts to the new extremum automatically amid changes in the simulated wave conditions. Moreover, no explicit knowledge of (future) wave excitation forces is required in the algorithm, which implies that the model-free ES can be used as a causal controller for WECs. Our results demonstrate that the continuous-time and model-free ES method achieves the optimum within a single simulation, which is in contrast to evolution-based optimization strategies that typically require a large number of (possibly expensive) function evaluations. This makes ES control optimization schemes suitable for nonlinear computational fluid dynamics simulations, where typically evolutionary strategies are used for performing black-box optimization.