The main focus of this thesis is to evaluate $kr(n,\delta )$, the minimal number of $r$-cliques in graphs with $n$ vertices and minimum degree~$\delta$. A fundamental result in Graph Theory states that a triangle-free graph of order $n$ has at most $n2/4$ edges. Hence, a triangle-free graph has minimum degree at most $n/2$, so if $k3(n,\delta )=0$ then $\delta \le n/2$. For $n/2\le \delta \le 4n/5$, I have evaluated $kr(n,\delta )$ and determined the structures of the extremal graphs. For $\delta \ge 4n/5$, I give a conjecture on $kr(n,\delta )$, as well as the structures of these extremal graphs. Moreover, I have proved various partial results that support this conjecture. Let $krreg(n,\delta )$ be the analogous version of $kr(n,\delta )$ for regular graphs. Notice that there exist $n$ and $\delta$ such that $kr(n,\delta )=0$ but $krreg(n,\delta )>0$. For example, a theorem of Andr{'a}sfai, Erd{\H{o}}s and S{'o}s states that any triangle-free graph of order $n$ with minimum degree greater than $2n/5$ must be bipartite. Hence $k3(n,\lfloor n/2\rfloor )=0$ but $k3reg(n,\lfloor n/2\rfloor )>0$ for $n$ odd. I have evaluated the exact value $k3reg(n,\delta )$ for $\delta$ between $2n/5+12n/5$ and $n/2$ and determined the structure of these extremal graphs. At the end of the thesis, I investigate a question in Ramsey Theory. The Ramsey number $Rk(G)$ of a graph $G$ is the minimum number $N$, such that any edge colouring of $KN$ with $k$ colours contains a monochromatic copy of $G$. The constrained Ramsey number $f(G,T)$ of two graphs $G$ and $T$ is the minimum number $N$ such that any edge colouring of $KN$ with any number of colours contains a monochromatic copy of $G$ or a rainbow copy of $T$. It turns out that these two quantities are closely related when $T$ is a matching. Namely, for almost all graphs $G$, $f(G,tK2)=Rt-1(G)$ for $t\ge 2$.