  Denote by $tC_\ell$ the disjoint union of $t$ cycles of length $\ell$. Let
$ex(n,F)$ and $spex(n,F)$ be the maximum size and spectral radius over all
$n$-vertex $F$-free graphs, respectively. In this paper, we shall pay attention
to the study of both $ex(n,tC_\ell)$ and $spex(n,tC_\ell)$. On the one hand, we
determine $ex(n,tC_{2\ell+1})$ and characterize the extremal graph for any
integers $t,\ell$ and $n\ge f(t,\ell)$, where $f(t,\ell)=O(t\ell^2)$. This
generalizes the result on $ex(n,tC_3)$ of Erd\H{o}s [Arch. Math. 13 (1962)
222--227] as well as the research on $ex(n,C_{2\ell+1})$ of F\"{u}redi and
Gunderson [Combin. Probab. Comput. 24 (2015) 641--645]. On the other hand, we
focus on the spectral Tur\'{a}n-type function $spex(n,tC_{\ell})$, and
determine the extremal graph for any fixed $t,\ell$ and large enough $n$. Our
results not only extend some classic spectral extremal results on triangles,
quadrilaterals and general odd cycles due to Nikiforov, but also develop the
famous spectral even cycle conjecture proposed by Nikiforov (2010) and
confirmed by Cioab\u{a}, Desai and Tait (2022).
