The objects of the talk are the commutative group schemes of finite type over a field. Their structure is well-understood in the case of linear algebraic groups over the field of complex numbers. But the general case is much more complicated, already for two-dimensional smooth connected algebraic groups. The talk will first discuss some known structure theorems. Then we will present old and new results on the (abelian) category of commutative algebraic groups, by developing the analogy with the category of coherent sheaves on an algebraic variety.