We introduce and analyse infinite dimensional Wishart processes taking values in the cone $S^+_1(H)$ of positive self-adjoint trace class operators on a separable real Hilbert space $H$. Our main result gives necessary and sufficient conditions for their existence, showing that these processes are necessarily of fixed finite rank almost surely, but they are not confined to a finite-dimensional subspace of $S^+_1(H)$. By providing explicit solutions to operator valued Riccati equations, we prove that their Fourier-Laplace transform is exponentially affine in the initial value. As a corollary, we obtain uniqueness in law as well as the Markov property. We actually show the explicit form of the Fourier-Laplace transform for a wide parameter regime, thereby also extending what is known in the finite-dimensional setting. Finally, under minor conditions on the parameters we prove the Feller property with respect to a slight refinement of the weak-$*$-topology on $S_1^+(H)$. Applications of our results range from tractable infinite-dimensional covariance modelling to the analysis of the limit spectrum of large random matrices.