Full waveform inversion (FWI) commonly stands for the state-of-the-art approach for imaging subsurface structures and physical parameters, however, its implementation usually faces great challenges, such as building a good initial model to escape from local minima, and evaluating the uncertainty of inversion results. In this paper, we propose the implicit full waveform inversion (IFWI) algorithm using continuously and implicitly defined deep neural representations. Compared to FWI, which is sensitive to the initial model, IFWI benefits from the increased degrees of freedom with deep learning optimization, thus allowing to start from a random initialization, which greatly reduces the risk of non-uniqueness and being trapped in local minima. Both theoretical and experimental analyses indicates that, given a random initial model, IFWI is able to converge to the global minimum and produce a high-resolution image of subsurface with fine structures. In addition, uncertainty analysis of IFWI can be easily performed by approximating Bayesian inference with various deep learning approaches, which is analyzed in this paper by adding dropout neurons. Furthermore, IFWI has a certain degree of robustness and strong generalization ability that are exemplified in the experiments of various 2D geological models. With proper setup, IFWI can also be well suited for multi-scale joint geophysical inversion.