Abstract: The Riemann problem for a class of flux-perturbation Euler equations with general pressure is solved, and five kinds of Riemann solutions of different structures are obtained. It is shown that when the three-parameter flux perturbations including pressure disappear, any Riemann solution containing two shock waves converges to a delta-shock solution of the zero-pressure flow; any Riemann solution including two rarefaction waves converges to the vacuum solution of the zero-pressure flow. It is also proved that when the two-parameter flux perturbations including pressure disappear, any two-shock Riemann solution satisfying certain initial values converges to a delta-shock solution of a type of Chaplygin gas equations. Finally, the formation processes of the delta-shocks and vacuum states are simulated numerically.