In this talk we revisit the complete eigenstructure problem of polynomial matrices of degree $d$ : $$ P(\lambda)=P_0+P_1\lambda + \ldots + P_d \lambda^d, \quad P_i\in \Re^{m\times n} $$ and the problem of finding equivalent characterizations of this eigenstructure via polynomial matrices of lower degree. It is well known that the Smith normal form of $P(\lambda)$ plays a crucial role for its zero structure but it is less well known how to describe the so-called left and right null space structures of $P(\lambda)$. This is normally done via the notion of minimal bases of rational vector spaces, which were quite popular in the 1970's in systems and control theory. If a minimal basis is $n_1$-dimensional and is arranged as the rows of a $n_1\times n$ polynomial matrix $Z_1(\lambda)$ then the so-called dual space is $n_2$-dimensional and is spanned by the rows of a $n_2\times n$ polynomial matrix $Z_2(\lambda)$ such that $Z_1(\lambda)Z_2(\lambda)^T=0$ and whose rows form also a minimal basis, i.e. these two bases span each others null space and their dimensions add up to the full dimension $n_1+n_2=n$. The matrices $Z_1(\lambda)$ and $Z_2(\lambda)$ are called dual minimal bases and play a crucial role in the problems we address in this talk, which is why we first look at their properties such as degree constraints and smoothness under perturbations. In the second part of the talk we show the importance of these dual minimal bases in three applications : the construction and perturbation theory of linearizations of a given polynomial matrix $P(\lambda)$, the construction of $\ell$-ifications of $P(\lambda)$ and the inverse problem of polynomial matrices with given eigenstructure.