  Let $G_1,...,G_n$ be graphs on the same vertex set of size $n$, each graph
with minimum degree $\delta(G_i)\ge n/2$. A recent conjecture of Aharoni
asserts that there exists a rainbow Hamiltonian cycle i.e. a cycle with edge
set $\{e_1,...,e_n\}$ such that $e_i\in E(G_i)$ for $1\leq i \leq n$. This can
be viewed as a rainbow version of the well-known Dirac theorem. In this paper,
we prove this conjecture asymptotically by showing that for every
$\varepsilon>0$, there exists an integer $N>0$, such that when $n>N$ for any
graphs $G_1,...,G_n$ on the same vertex set of size $n$ with $\delta(G_i)\ge
(\frac{1}{2}+\varepsilon)n$, there exists a rainbow Hamiltonian cycle. Our main
tool is the absorption technique. Additionally, we prove that with
$\delta(G_i)\geq \frac{n+1}{2}$ for each $i$, one can find rainbow cycles of
length $3,...,n-1$.
