  In 1970, Walkup completely described the set of $f$-vectors for the four
3-manifolds $S^3$, $S^2 twist S^1$, $S^2 \times S^1$, and $RP^3$. We improve
one of Walkup's main restricting inequalities on the set of $f$-vectors of
3-manifolds. As a consequence of a bound by Novik and Swartz, we also derive a
new lower bound on the number of vertices that are needed for a combinatorial
$d$-manifold in terms of its $\beta_1$-coefficient, which partially settles a
conjecture of K\"uhnel. Enumerative results and a search for small
triangulations with bistellar flips allow us, in combination with the new
bounds, to completely determine the set of $f$-vectors for twenty further
3-manifolds, that is, for the connected sums of sphere bundles $(S^2 \times
S^1)^{# k}$ and twisted sphere bundles $(S^2 twist S^1)^{# k}$, where
$k=2,3,4,5,6,7,8,10,11,14$. For many more 3-manifolds of different geometric
types we provide small triangulations and a partial description of their set of
$f$-vectors. Moreover, we show that the 3-manifold $RP^3 # RP^3$ has (at least)
two different minimal $g$-vectors.
