A bounded linear operator $T: {\mathcal H}\rightarrow {\mathcal H}$ is a $C$-normal operator if there exists a conjugation $C$ on ${\mathcal H}$ such that $[CT, (CT)^{\ast}]=0$ where $[R,S]:=RS-SR.$ We study several properties of $C$-normal operators. In particular, we show that $T-\lambda I$ is $C$-normal for all $\lambda\in{\mathbb C}$ if and only if $T$ is a complex symmetric operator with the conjugation $C$. In addition, we prove that if $T$ is $C$-normal, then the following statements are equivalent; (i) $T$ is normal, (ii) $T$ is quasinormal, (iii) $T$ is hyponormal, (iv) $T$ is $p$-hyponormal for $0