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In this thesis, we investigate the Cauchy problem for the quasilinear stochastic evolution equation \begin{equation*} \begin{cases} u(t)=[{-}A(u(t))u(t)+f(t)]\operatorname{dt}+B(u(t))dW(t),\quad t\in [0,T],\\ u(0)=u_0 \end{cases} \end{equation*} in a Banach space $ X. $ In the first part of the thesis, we concentrate on the parabolic situation, i.e. we assume that $ {-}A(u(t)) $ is for every $ t $ a generator of an analytic semigroup and that $ A(u(t)) $ has a bounded $ H^{\infty} $-calculus. Under a local Lipschitz assumption on $ u\mapsto A(u) $ we prove existence and uniqueness of a local strong solution up to a maximal stopping time that can be characterised by a blow-up alternative. We apply our local well-posedness result to a second order parabolic partial differential equation on $ \mathbb{R} ^d $, to a generalised Navier-Stokes equation describing non-Newtonian fluids and to a convection-diffusion equation on a bounded domain with Dirichlet, Neumann or mixed boudary conditions. In the last situation, we can even show that the solution exists globally. In the second part of the thesis, we go to a special hyperbolic situation. We look at a Maxwell equation on a domain $ D $ with perfect conductor boundary condition in chiral media with a nonlinear retarded material law, i.e. we consider $$ A(u)u(t)=-Mu(t)+|u(t)|^qu(t)-\int_{0}^{t}G(t-s)u(s)\operatorname{ds}. $$ Here, $ M(u_1,u_2)=(\operatorname{curl} u_2,-\operatorname{curl} u_1)^T $ is the Maxwell operator on $ L^{2}(D)^3\times L^{2}(D)^3 $. To solve this equation we apply a refined version of the monotonicity approach using the spectral multipliers of the Hodge-Laplace operator, which is a componentwise Laplace operator with boundary conditions comparable to those of $ M^2. $ We show existence and uniqueness of a weak solution $ u $ in the sense of partial differential equations and under stronger assumptions we prove that $ u $ is a strong solution, i.e. $ Mu(t,x) $ exists almost surely for almost all $ t\in[0,T] $ and $ x\in D $.