The purpose of this paper is to study the free Ornstein--Uhlenbeck processes in finite von Neumann algebras, formulated in Voiculescu's free probability. A~probability measure on \({\mathbb R}\) is free self-decomposable if and only if it is the limit distribution of a free Ornstein--Uhlenbeck process driven by a free Lévy process. The author shows first that a free self-decomposable probability measure on \({\mathbb R}\) can be realized as the distribution of a stationary free Ornstein--Uhlenbeck process driven by a free Lévy process. The author introduces free periodic Ornstein--Uhlenbeck processes driven by free Lévy processes, and a characterization of the stationary distribution of a free periodic Ornstein--Uhlenbeck process is given in terms of its Lévy measure. These results are parallel to the recent results on classical periodic Ornstein--Uhlenbeck processes, see, e.g., \textit{J.\,Petersen} [J.~Appl.\ Probab.\ 39, No.\,4, 748--763 (2002; Zbl 1023.60041)]. Finally, the notion of a free fractional Brownian motion is introduced. It is shown that the free stochastic differential equation driven by fractional free Brownian motion has a unique solution, called a fractional free Ornstein--Uhlenbeck process. Examples of free fractional Brownian motion are also given in terms of creation and anihilation operators on full Fock spaces. For other related works, see \textit{P.\,Biane}, and \textit{R.\,Speicher} [Ann.\ Inst.\ Henri Poincaré, Probab.\ Stat.\ 37, No.\,5, 581--606 (2001; Zbl 1020.46018)] for free Ornstein--Uhlenbeck processes driven by a free Brownian motion, and \textit{O.\,E.\,Barndorff-Nielsen} and \textit{S.\,Thorbjørnsen} [Proc.\ Natl.\ Acad.\ Sci.\ USA 99, No.\,26, 16576--16580 (2002; Zbl 1063.46052)] for free Ornstein--Uhlenbeck processes driven by free Lévy processes. For recent results on fractional Ornstein--Uhlenbeck processes driven by fractional Brownian motion in the classical probability, the reader may consult \textit{P.\,Cheridito, H.\,Kawaguchi}, and \textit{M.\,Maejima} [Electron.\ J.\ Probab.\ 8, Paper No.\,3 (2003; Zbl 1065.60033)].