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We investigate the function $ex(n,H,\\mathcal F)$, when $H$ and members of $\\mathcal F$ are cycles. Let $C_k$ denote the cycle of length $k$ and let $\\mathscr C_k=\\{C_3,C_4,\\ldots,C_k\\}$. Some of our main results are the following.\n  (i) We show that $ex(n, C_{2l}, C_{2k}) = \\Theta(n^l)$ for any $l, k \\ge 2$. 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