One of the central and oldest problems in combinatorics is : Can you find a collection $S$ of $q$-subsets from an $n$-element set $X$ so that every $r$-subset of $X$ is included in precisely $t$ sets in the collection ? A collection $S$ of this kind is called a design of parameters $(n,q,r,t)$, a special interest is the case $t=1$, and in this case $S$ is called a Steiner system. It was conjectured that a design of parameters $(n,q,r, t)$ exists for every $q$, $r$ and $t$ if $n$ is sufficiently large and is admissible, namely if some necessary simple divisibility conditions are satisfied. Until recently the known constructions came very far from this. Here is a brief history of the problem : The existence of designs and Steiner systems was asked by Plücker (1835), Kirkman (1846) and Steiner (1853). For $r=2$ which was of special interest, Richard Wilson (1972–1975) proved their existence for large enough admissible values of $n$. Rödl (1985) proved the existence of approximate objects (the property holds for $(1-o(1))$ $r$-subsets of $X$), thus answering a conjecture by Erdös and Hanani. Teirlink (1987) proved their existence for infinitely many values of $n$ when $r$ and $q$ are arbitrary and $t$ is a certain large number depending on $q$ and $r$ but not on $n$. (His construction also does not have repeated blocks.) Keevash (2014) proved the existence of Steiner systems for all but finitely many admissible values of $n$ for every $q$ and $r$. He uses a new method referred to as Randomized Algebraic Constructions. In the lecture I will describe the problem and its history and will try to explain some ingredients in the methods by Wilson, Rödl and Keevash.