The paper concerns multiparameter eigenvalue problems (1) \(A(\lambda,\nu,\mu)u=0\), \(\lambda\),\(\nu\in C\), \(\mu \in C^ m\), \(m\in N\), when A(\(\lambda\),\(\nu\),\(\mu)\) is a linear Fredholm operator with index 0 between Banach spaces E, F which is bounded, holomorphic on a region \(g\subset C^{m+2}\). The isolated eigenvalues \(\lambda =\lambda (\nu,\mu)\) of (1) are considered as functions of \(\nu\) and \(\mu\). Approximations of (1) by a family of analogous problems (2) \(A_ h(\lambda,\nu,\mu)u_ h=0\), \(h\in H\), are introduced, and, under suitable conditions, the convergence of the eigenvalues \(\lambda_ h(\nu,\mu)\) of (2) to those of (1) is proved and the stability of the eigenvalue manifolds under the approximation is studied. Then continuation methods are considered for the computation of eigenvalue curves and manifolds of limit points. For both continuation problems algebraic bifurcation problems are established to switch branches at bifurcation points. The results are applied to two problems from mechanics, namely, the buckling of a column and Mathieu's equation with damping.