An abelian variety $A$ is an algebraic variety which is also an algebraic group. 1-dimensional abelian varieties are all elliptic curves, and it is a celebrated theorem of Mordell that the rank of the algebraic group $E(\mathbb{Q})$ of rational points of an elliptic curve $E$ is finitely generated. This was later generalized by Weil; that is, for an abelian variety $A$ and any number field $K$, the group $A(K)$ of $K$-rational points of $A$ is a finitely generated abelian group. The rank of the group $A(\mathbb{Q})$ is called the Mordell-Weil rank of $A$.
The question of whether the rank of elliptic curves is absolutely bounded is a difficult one, and the consensus on whether it should have a positive or negative answer shifts with time. When I started graduate school it seemed at the time that the consensus was that the rank of elliptic curves ought to be unbounded; that is, for any real number $M$ one can find an elliptic curve $E/\mathbb{Q}$ whose Mordell-Weil rank exceeds $M$. It seems that this consensus have swung the other way in very recent times, due to various heuristics given by convincing random models. There are many mathematicians who contributed to this work (too many to properly name and attribute here).
If we extend the question to include all abelian varieties, then a naive version of this question is trivial: given any real number $M$, one can find an abelian variety $A/\mathbb{Q}$ whose Mordell-Weil rank exceeds $M$. For example, one has the following theorem of Coleman:
Theorem (R. Coleman): Let $f(x)$ be a monic polynomial with integer coefficients and degree $k$. Suppose that there exists $p > 2$ such that $f(x) \equiv x^k \pmod{p}$ and that $f(x)$ has at least $(k+1)/2$ distinct roots over the integers. Then the Mordell-Weil rank of the hyperelliptic curve $$\displaystyle y^2 = f(x) + 1$$ is at least $(k-1)/2$.
In particular, this shows that the Jacobian variety of the curve above has rank at least $(k-1)/2$. Choosing $k$ to be arbitrarily large, one sees that the rank of abelian varieties cannot be absolutely bounded.
However, a more natural question is to ask, given a positive integer $g$, whether there exists a positive number $N(g)$ such that all but finitely many abelian varieties (up to isomorphism) of dimension $g$ has Mordell-Weil rank bounded by $N(g)$. In this formulation, the above heuristics used to justify that the rank of elliptic curves ought to be absolutely bounded seems to show that $N(1) = 21$.
Is there a consensus on whether $N(g)$ should exist for $g \geq 2$? If so, what is conjectured about the growth of $N(g)$ with respect to $g$, as $g \rightarrow \infty$?
