A single change covering design (SCCD) is a sequence of $b$ $k$-sets, called blocks, of a $V$-set in which exactly one element differs between consecutive blocks and every $s$-set of $V$ is in some block. We will review the literature and discuss several recursive constructions which completely solve the existence of SCCD for $k=3,4$ and partially for $k=5$. We will examine optimizations on exhaustive search techniques. We determined that there are 313 unique circular SCCD(12,4,2,22). We determine a recursion for $s=3$ and general $k$ using expansion sets. A double change covering design (DCCD) is similarly defined but consecutive blocks differ by two elements. We will completely solve the existence of tight DCCD ~$k=3,s=2$. We give several constructions using recursion and algebraic difference methods. This provides us with constructions for circular DCCD$(4k-2,2k,2,2k-1)$, circular DCCD$(4k-1,2k+1,2,2k-1)$ and circular DCCD$(4k-5k,2,4k-2)$ exists. We also find some other circular DCCD ~with given $v$ and $k$.