  In this paper, the fully parabolic Keller-Segel system \begin{equation}
\left\{ \begin{array}{llc} u_t=\Delta u-\nabla\cdot(u\nabla v), &(x,t)\in
\Omega\times (0,T),\\ v_t=\Delta v-v+u, &(x,t)\in\Omega\times (0,T),\\
\end{array} \right.
  \qquad \qquad (\star) \end{equation} is considered under Neumann boundary
conditions in a bounded domain $\Omega\subset\mathbb{R}^n$ with smooth
boundary, where $n\ge 2$. We derive a smallness condition on the initial data
in optimal Lebesgue spaces which ensure global boundedness and large time
convergence. More precisely, we shall show that one can find $\varepsilon_0>0$
such that for all suitably regular initial data $(u_0,v_0)$ satisfying
$\|u_0\|_{L^{\frac{n}{2}}(\Omega)}<\varepsilon_0$ and $\|\nabla
v_0\|_{L^{n}(\Omega)}<\varepsilon_0$, the above problem possesses a global
classical solution which is bounded and approaches the constant steady state
$(m,m)$ with $m:=\frac{1}{|\Omega|}\int_{\Omega} u_0$. Our approach allows us
to furthermore study a general chemotaxis system with rotational sensitivity in
dimension 2, which is lacking the natural energy structure associated with
($\star$). For such systems, we prove a global existence and boundedness result
under corresponding smallness conditions on the initially present total mass of
cells and the chemical gradient.
