The paper studies the notion of Stepanov almost periodicity (or S 2 -almost periodicity) for stochastic processes, which is weaker than the notion of quadratic-mean almost periodicity. Next, we make extensive use of the so-called Acquistapace and Terreni conditions to prove the existence and uniqueness of a Stepanov (quadratic-mean) almost periodic solution to a class of nonautonomous stochastic evolution equations on a separable real Hilbert space. Our abstract results will then be applied to

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... tudy Stepanov (quadraticmean) almost periodic solutions to a class of n-dimensional stochastic parabolic partial differential equations. 2000 Mathematics Subject Classification. 34K14 , 60H10, 35B15, 34F05.