  A noise reinforced Brownian motion is a centered Gaussian process $\hat
B=(\hat B(t))_{t\geq 0}$ with covariance $E(\hat B(t)\hat
B(s))=(1-2p)^{-1}t^ps^{1-p} \quad \text{for} \quad 0\leq s \leq t,$ where
$p\in(0,1/2)$ is a reinforcement parameter. Our main purpose is to establish a
version of Donsker's invariance principle for a large family of step-reinforced
random walks in the diffusive regime, and more specifically, to show that $\hat
B$ arises as the universal scaling limit of the former. This extends known
results on the asymptotic behavior of the so-called elephant random walk.
