In this paper, we construct $q$ -ary two-deletion correcting codes and burst-deletion correcting codes, where $q\geq 2$ is an even integer. For two-deletion codes, our construction has redundancy $5\log n+O(\log q\log \log n)$ and has encoding complexity near-linear in $n$ , where $n$ is the length of the message sequences. For burst-deletion codes, we first present a construction of binary codes with redundancy $\log n+9\log \log n+\gamma _{t}+o(\log \log n)$ bits $(\gamma _{t}$ is a constant that depends only on $t$ ) and capable of correcting a burst of at most $t$ deletions, which improves the Lenz-Polyanskii Construction (ISIT 2020). Then we give a construction of $q$ -ary codes with redundancy $\log n+(8\log q+9)\log \log n+\gamma _{t}+o(\log \log n)$ bits and capable of correcting a burst of at most $t$ deletions.