the lecture will discuss recent joint work with costante bellettini at ucl. a main outcome of the work is a proof that for any closed riemannian manifold $n$ of dimension $n \geq 3$ and any non-negative (or non-positive) lipschitz function $g$ on $n$, there is a boundaryless $c^{2}$ hypersurface $m \subset n$ whose scalar mean curvature is prescribed by $g.$ more precisely, the hypersurface $m$ is the image of a quasi-embedding $\iota$ (of class $c^{2}$) admitting a global unit normal $\nu$ such that the mean curvature of $\iota$ at every point $x$ is $g(\iota(x))\nu(x)$. here a 'quasi-embedding' is an immersion such that any point of its image near which the image is not embedded has an ambient neighborhood in which the image is the union of two $c^{2}$ embedded disks with each disk lying on one side of the other (so that any self-intersection is tangential). if $n \geq 7$, the singular set $\overline{m} \setminus m$ may be non-empty, but has hausdorff dimension no greater than $n-7$. an important special case is the existence of a cmc hypersurface for any prescribed value of mean curvature. the method of proof is pde theoretic. it utilises the elliptic and parabolic allen-cahn equations on $n$, and brings to bear on the question elementary, and yet very effective, variational and gradient flow principles in semi-linear elliptic and parabolic pde theory--principles that serve as a conceptually and technically simpler alternative to the geometric measure theory machinery pioneered by almgren and pitts to prove existence of a minimal hypersurface. for regularity conclusions the method relies on a new varifold regularity theory, a ''black-box'' tool of independent interest (also joint work with bellettini). this theory provides multi-sheeted $c^{1, \alpha}$ regularity for mean-curvature-controlled codimension 1 integral varifolds $v$ near points where one tangent cone is a hyperplane of multiplicity $q \geq 2;$ this regularity holds whenever: (i) $v$ has no classical-singularities, i.e. no portion of $v$ is the union of three or more embedded hypersurfaces-with-boundary coming smoothly together along their common boundary, and (ii) the region where the mass density of $v$ is $< q$ is 'well-behaved' in a certain topological sense. a very important feature of this theory, crucial for its application to the allen--cahn method, is that $v$ is not assumed to be a critical point of a functional.