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An $r$-uniform hypergraph is called an $r$-graph. A hypergraph is linear ifevery two edges intersect in at most one vertex. Given a linear $r$-graph $H$and a positive integer $n$, the linear Tur\'an number $ex_L(n,H)$ is themaximum number of edges in a linear $r$-graph $G$ that does not contain $H$ asa subgraph. For each $\ell\geq 3$, let $C^r_\ell$ denote the $r$-uniform linearcycle of length $\ell$, which is an $r$-graph with edges $e_1,\ldots, e_\ell$such that $\forall i\in [\ell-1]$, $|e_i\cap e_{i+1}|=1$, $|e_\ell\cap e_1|=1$and $e_i\cap e_j=\emptyset$ for all other pairs $\{i,j\}, i\neq j$. For all$r\geq 3$ and $\ell\geq 3$, we show that there exist positive constants$c_{m,r}$ and $c'_{m,r}$, depending only $m$ and $r$, such that$ex_L(n,C^r_{2m})\leq c_{m,r} n^{1+\frac{1}{m}}$ and $ex_L(n,C^r_{2m+1})\leqc'_{m,r} n^{1+\frac{1}{m}}$. This answers a question of Kostochka, Mubayi, andVerstra\"ete. For even cycles, our result extends the result of Bondy andSimonovits on the Tur\'an numbers of even cycles to linear hypergraphs.  Using our results on linear Tur\'an numbers we also obtain bounds on thecycle-complete hypergraph Ramsey numbers. We show that there are positiveconstants $a_{m,r}$ and $b_{m,r}$, depending only on $m$ and $r$, such that$R(C^r_{2m}, K^r_t)\leq a_{m,r} (\frac{t}{\ln t})^\frac{m}{m-1}$ and$R(C^r_{2m+1}, K^r_t)\leq b_{m,r} t^\frac{m}{m-1}$.

Linear Turan numbers of r-uniform linear cycles and related Ramsey numbers