The main purpose of this paper is to analyze the stability of the implicit-explicit (IMEX) time-marching methods coupled with high order finite difference spatial discretization for solving the linear convection-diffusion and convection-dispersion equations in one dimension. Both Runge-Kutta and multistep IMEX methods are considered. Stability analysis is performed on the above mentioned schemes with uniform meshes and periodic boundary condition by the aid of the Fourier method. For the convection-diffusion equations, the result shows that the high order IMEX finite difference schemes are subject to the time step restriction $∆t ≤$ max{$τ_0, c∆x$}, where $τ_0$ is a positive constant proportional to the diffusion coefficient and c is the Courant number. For the convection-dispersion equations, we show that the IMEX finite difference schemes are stable under the standard CFL condition $∆t ≤ c∆x$. Numerical experiments are also given to verify the main results.