  We develop a theoretical framework for the analysis of stabilized cut finite
element methods for the Laplace-Beltrami operator on a manifold embedded in
$\mathbb{R}^d$ of arbitrary codimension. The method is based on using
continuous piecewise polynomials on a background mesh in the embedding space
for approximation together with a stabilizing form that ensures that the
resulting problem is stable. The discrete manifold is represented using a
triangulation which does not match the background mesh and does not need to be
shape-regular, which includes level set descriptions of codimension one
manifolds and the non-matching embedding of independently triangulated
manifolds as special cases. We identify abstract key assumptions on the
stabilizing form which allow us to prove a bound on the condition number of the
stiffness matrix and optimal order a priori estimates. The key assumptions are
verified for three different realizations of the stabilizing form including a
novel stabilization approach based on penalizing the surface normal gradient on
the background mesh. Finally, we present numerical results illustrating our
results for a curve and a surface embedded in $\mathbb{R}^3$.
