In this paper it is shown that Toeplitz operators on Bergman space form a dense subset of the space of all bounded linear operators, in the strong operator topology, and that their norm closure contains all compact operators. Further, the$C$^{*}-algebra generated by them does not contain all bounded operators, since all Toeplitz operators belong to the essential commutant of certain shift. The result holds in Bergman spaces$A$^{2}(Ω) for a wide class of plane domains Ω⊂$C$, and in Fock spaces$A$^{2}($C$^{N}),$N$≧1.