We study the asymptotic behavior of short cycles of random permutations with cycle weights. More specifically, on a specially constructed metric space whose elements encode all possible cycles, we consider a point process containing all information on cycles of a given random permutation on $\{1,\ldots,n\}$. The main result of the paper is the distributional convergence with respect to the vague topology of the above processes towards a Poisson point process as $n\to\infty$ for a wide range of cycle weights. As an application, we give several limit theorems for various statistics of cycles.