In this paper we consider weakly hyperbolic equations of higher orders in
arbitrary dimensions with time-dependent coefficients and lower order terms. We
prove the Gevrey well-posedness of the Cauchy problem under $C^k$-regularity of
coefficients of the principal part and natural Levi conditions on lower order
terms which may be only continuous. In the case of analytic coefficients in the
principal part we establish the $C^\infty$ well-posedness. The proofs are based
on using the quasi-symmetriser for the corresponding system. The main novelty
compared to the existing literature is the possibility to include lower order
terms to the equation as well as considering any space dimensions. We also give
results on the ultradistributional and distributional well-posedness of the
problem, and we look at new effects for equations with discontinuous lower
order terms.