The thesis consists of 3 parts. In the first part some problems from Extremal poset theory are studied, including the Diamond problem, which is one of the most investigated problems in this area, and give an improved bound. We also show that an induced P-free family has size $O(binom{n}{n/2})$, proving a conjecture of Katona, and Lu and Milans. In the second part of the thesis, we study problems in extremal graph theory, mainly concerning cycles of even length: We answer a question of Kuhn and Osthus concerning subgraphs of $C_{2k)$-free graphs. We also study Turán numbers of ordered even cycles and generalised Turán problems for even cycles. Moreover, we determine the asymptotic value of maximum possible number of edges in a $C_{2k+1}$-free graph containing no induced copy of $K_{s,t}$, answering a question of Loh, Tait, Timmons and Zhou. In the third part of the thesis, we study Hypergraph Turán problems. In particular, we study the Turán numbers of Berge cycles and Berge-$K_{2,t}$, among others.