  In this paper we consider integration and $L_2$-approximation for functions
over $\RR^s$ from weighted Hermite spaces. The first part of the paper is
devoted to a comparison of several weighted Hermite spaces that appear in
literature, which is interesting on its own. Then we study tractability of the
integration and $L_2$-approximation problem for the introduced Hermite spaces,
which describes the growth rate of the information complexity when the error
threshold $\varepsilon$ tends to 0 and the problem dimension $s$ grows to
infinity. Our main results are characterizations of tractability in terms of
the involved weights, which model the importance of the successive coordinate
directions for functions from the weighted Hermite spaces.
