  Lax pairs are a useful tool in finding conserved quantities of some dynamical
systems. In this expository article, we give a motivated introduction to the
idea of a Lax pair of matrices $(L,A)$, first for mechanical systems such as
the linear harmonic oscillator, Toda chain, Eulerian rigid body and the
Rajeev-Ranken model. This is then extended to Lax operators for one-dimensional
field theories such as the linear wave and KdV equations and reformulated as a
zero curvature representation via a $(U,V)$ pair which is illustrated using the
nonlinear Schr\"odinger equation. The key idea is that of realizing a
(possibly) nonlinear evolution equation as a compatibility condition between a
pair of linear equations. The latter could be an eigenvalue problem for the Lax
operator $L$ and a linear evolution equation generated by $A$, for the
corresponding eigenfunction. Alternatively, they could be the first order
linear system stating the covariant constancy of an arbitrary vector with
respect to the 1+1 dimensional gauge potential $(V,U)$. The compatibility
conditions are then either the Lax equation $\dot L = [L, A]$ or the flatness
condition $U_t - V_x + [U, V] = 0$ for the corresponding gauge potential. The
conserved quantities then follow from the isospectrality of the Lax and
monodromy matrices.
