In this work we consider the three-dimensional solvable Lie group denoted by $Sol_{3}$, equipped with any left-invariant metric, either Lorentzian or Riemannian. The existence of non-trivial (i.e., not Einstein) Ricci solitons on both Lorentzian and Riemannian three-dimensional solvable Lie group $Sol_{3}$ is proved. Moreover, we show that they are not gradient Ricci solitons.