A central problem of extremal combinatorics is
to determine the Tur\'an number of a given
$r$-uniform hypergraph $\mathcal{F}$, i.e. the
maximum number of edges in an $r$-uniform hypergraph
on $n$ vertices that does not contain a copy of
$\mathcal{F}$. Since the problem was introduced over
sixty years ago, it has only been solved for relatively
few hypergraphs $\mathcal{F}$. Many of these results
were found very recently by means of the stability method,
which has brought new life to research in a challenging
area. However, this method only has the potential to
solve the problem when the extremal configuration is unique,
so in other cases we need new techniques.

In this talk we will discuss some methods for Tur\'an problems
due to various authors, that incorporate some algebraic
ideas. In particular we will present two new results: one
gives bounds for the Tur\'an numbers of projective geometries,
and another gives a general bound for the Tur\'an number
of a hypergraph in terms of the number of edges that it
contains.