This study was conducted to find a catalyst system which has high conversion and selectivity for the oxidative coupling of methane to produce ethane and ethylene. Various catalysts were tested in a fixed bed reactor ar $750^{circ}C$, 1 atm, and the feed ratio($CH_4/O_2$) of 2/1. Under the reaction condition, 10wt%$PbSO_4/MgO$ catalyst showed the highest catalytic activity : methane conversion, $C_2$ selectivity and yield were 50, 40 and 20%, respectively. Catalysts containing sulfate compounds, 10wt%$PbSO_4/MgO$, 10wt%$MgSO_4/MgO$ and $Na_2SO_4/MgO$ revealed a moderate methane conversions such as 38, 50 and 50%, respectively and low $C_2$ selectivities such as 18, 5 and 9%, respectively. Catalysts containing carbonate compounds, 10wt%$PbCO_3/MgO$, 10wt%$Li_2CO_3/MgO$ and $NaCO_3/MgO$, also showed a moderate methane conversions such as 64, 44 and 51%, respectively and low $C_2$ selectivities such as 5, 6 and 2%, respectively. With the existence of chlorine and mercury, $C_2$ selectivity was decreased.