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We collect, scattered through literature, as well as we prove some newproperties of two Markov processes that in many ways resemble Wiener andOrnstein--Uhlenbeck processes. Although processes considered in this paper weredefined either in non-commutative probability context or through quadraticharnesses we define them once more as so to say 'continuous time 'generalization of a simple, symmetric, discrete time process satisfying simpleconditions imposed on the form of its first two conditional moments. The finitedimensional distributions of the first one (say X=(X_{t})_{t\geq0} calledq-Wiener) depends on one parameter q\in(-1,1] and of the second one (sayY=(Y_{t})_{t\inR} called ({\alpha},q)- Ornstein--Uhlenbeck) on two parameters({\alpha},q)\in(0,\infty)\times(-1,1]. The first one resembles Wiener processin the sense that for q=1 it is Wiener process but also that for |q|<1 and\foralln\geq1: t^{n/2}H_{n}(X_{t}/\surdt|q), where (H_{n})_{n\geq0} are the socalled q-Hermite polynomials, are martingales. It does not have however neitherindependent increments not allows continuous sample path modification. Thesecond one resembles Ornstein--Uhlenbeck process. For q=1 it is a classical OUprocess. For |q|<1 it is also stationary with correlation function equal toexp(-{\alpha}|t-s|) and has many properties resembling those of its classicalversion. We think that these process are fascinating objects to study posingmany interesting, open questions.Comment: 25 page

$q$-Wiener and $(\alpha,q)$-Ornstein--Uhlenbeck Processes. A Generalization of Known Processes