We study the gap probability of finding no eigenvalues near the cusp singularity in random matrix ensembles. It was known that the cusp singularity leads to a new universal determinantal process characterized by the Pearcey kernel. By studying the Fredholm determinant of the Pearcey kernel, we establish an integral representation of the gap probability in terms of the Hamiltonian of a system of nonlinear differential equations. The large gap asymptotics are then derived in both the Pearcey process and the thinned Pearcey process where each particle in the process is removed independently with certain probability. The hard edge Pearcey process and higher order Airy processes will also be discussed in the talk.