The fundamental idea of Laser Wakefield Acceleration (LWFA) is reviewed. An ultrafast intense laser pulse drives coherent wakefields of relativistic amplitude with the high phase velocity robustly supported by the plasma. The structures of wakes and sheaths in plasma are contrasted. While the large amplitude of wakefields involves collective resonant oscillations of the eigenmode of the entire plasma electrons, the wake phase velocity ~ c and ultrafastness of the laser pulse introduce the wake stability and rigidity. When the phase velocity gets smaller, wakefields turn into sheaths. When we deploy laser ion acceleration or high density LWFA in which the phase velocity of plasma excitation is low, we encounter the sheath dynamics. A large number of worldwide experiments show a rapid progress of this concept realization toward both the high energy accelerator prospect and broad applications. The strong interest in this has driven novel laser technologies, including the Chirped Pulse Amplification, the Thin Film Compression (TFC), the Coherent Amplification Network, and the Relativistic Compression (RC). These in turn have created a conglomerate of novel science and technology with LWFA to form a new genre of high field science with many parameters of merit in this field increasing exponentially lately. Applications such as ion acceleration, X-ray free electron laser, electron and ion cancer therapy are discussed. A new avenue of LWFA using nanomaterials is also emerging, adopting X-ray laser using the above TFC and RC. Meanwhile, we find evidence that the Mother Nature spontaneously created wakefields that accelerate electrons and ions to very high energies.