The classical von Neumann bicommutant theorem states that if \(B\) is a unital self-adjoint closed subalgebra of operators on a Hilbert space, then the bicommutant of \(B\) coincides with the closure of \(B\) in the weak operator topology. This theorem has no direct counterpart for algebras of operators on a Banach space. In the present paper the authors study versions of the bicommutant theorem for algebras of multiplication operators on \(L^{p}\)-spaces. Let \(\mu\) be a \(\sigma\)-finite measure space (weaker assumptions are sufficient) and let \(A\) be a subalgebra of \(L^{\infty}(\mu)\) containing the constants. Each \(\varphi\in L^{\infty}(\mu)\) operates on each \(L^{p}(\mu)\)-space via \(f\mapsto \varphi f\); denote \(M_{p}(A)= \{f\mapsto \varphi f: \varphi\in A\}\subset L(L^{p}(\mu))\). The main results of the paper are as follows. Theorem A: If \(1\leq p<\infty\) and the scalars are real, then the bicommutant of \(M_{p}(A)\) coincides with the weak operator closure of \(M_{p}(A)\), or, equivalently, with \(M_{p}(w^{*}\text{-clos}(A))\). Theorem B: If \( p=\infty\) and the scalars are real, then the bicommutant of \(M_{\infty}(A)\) coincides with \(M_{\infty}(R(A))\), where \(R(A)\) denotes the Dedekind closure of \(A\) in the Banach lattice \(L^{\infty}_{\mathbb R}(\mu)\). For example, if \(A=C[0,1]\subset L^{\infty}[0,1]\) with respect to Lebesgue measure, then \(R(A)\) consists of those equivalence classes in \(L^{\infty}[0,1]\) having a Riemann integrable representative. Similar results are valid for \(L^{p}\)-spaces over \(\mathbb C\).