In this talk, we first introduce several time-stepping and time-parallel time integration schemes. We then present a family of block implicit methods (BIM), which are defined by a tableau consisting of two matrices and two vectors. It is shown that fully implicit Runge-Kutta schemes, along with other time-parallel algorithms derived from classical time-stepping schemes (e.g., backward Euler, Crank-Nicolson, and BDF methods), are special cases of BIM. Further, we prove that the traditional finite element theory for parabolic problems discretized by the backward Euler or Crank-Nicolson schemes can also be extended for BIM. Several parallel preconditioners algorithms are also presented for solving the resulting large sparse linear system of equations. Finally, numerical experiments carried out on a parallel computer with thousands of processors confirm the optimality and scalability of the method.