The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system consists of interactions of four planar elementary waves. Different from polytropic gas, all of them are contact discontinuities due to the system is full linear degenerate, i.e., the three eigenvalues of the system are linear degenerate. They include compressive one ($S^{\pm}$), rarefactive one ($R^{\pm}$) and slip lines ($J^{\pm}$). We still call $S^{\pm}$ as shock and $R^{\pm}$ as rarefaction wave. According to different combinations of four elementary waves, one delivers a complete classification to the problem. It contains 14 cases in all. The Riemann solutions are self-similar, and the flow is transonic in the self-similar plane ($x/t,y/t$). The boundaries of the interaction domains are obtained. Solutions in supersonic domains are constructed in no $J$ cases. While in the rest cases, the structure of solutions are conjectured except for the case $2J^{+}+2J^{-}$. Especially, delta waves and simple waves appear in some cases. The Dirichlet boundary value problems in subsonic domains or the boundary value problems for transonic flow are formed case by case. The domains are convex for two cases, and non-convex for the rest cases. The boundaries of the domains are composed of sonic curves and/or slip lines.