Large-scale numerical simulations are essential to the design of plasma-based accelerators and laser-plasma interations for ultra-high intensity (UHI) physics. The electromagnetic Particle-In-Cell (PIC) approach is the method of choice for self-consistent simulations, as it is based on first principles, and captures all kinetic effects, and also scales easily (for uniform plasmas) to many cores on supercomputers. The standard PIC algorithm relies on second-order finite-difference discretizations of the Maxwell and Newton-Lorentz equations. We present here novel PIC formulations, based on the use of very high-order pseudo-spectral Maxwell solvers, which enable near-total elimination of the numerical Cherenkov instability and increased accuracy over the standard PIC method. We also discuss the latest implementations in the PIC modules Warp-PICSAR and FBPIC on the Intel Xeon Phi and GPU architectures. Examples of applications are summarized on the simulation of laser-plasma accelerators and high-harmonic generation with plasma mirrors.