  This is an elementary and self--contained review of twistor theory as a
geometric tool for solving non-linear differential equations. Solutions to
soliton equations like KdV, Tzitzeica, integrable chiral model, BPS monopole or
Sine-Gordon arise from holomorphic vector bundles over $T\CP^1$. A different
framework is provided for the dispersionless analogues of soliton equations,
like dispersionless KP or $SU(\infty)$ Toda system in 2+1 dimensions. Their
solutions correspond to deformations of (parts of) $T\CP^1$, and ultimately to
Einstein--Weyl curved geometries generalising the flat Minkowski space. A
number of exercises is included and the necessary facts about vector bundles
over the Riemann sphere are summarised in the Appendix.
