As J. Dieudonné eloquently pointed out in Chapter A V of *A panorama of pure mathematics (as seen by N. Bourbaki)*, this is an extremely broad question:

The theory of partial differential equations has been studied incessantly
for more than two centuries. By reason of its permanent symbiosis with
almost all parts of physics, as well as its ever closer connections with many other branches of mathematics, it is one of the largest and most diverse regions of present-day mathematics, and the vastness of its bibliography defies the imagination.

For a long time, the theory of ordinary differential equations served more
or less consciously as a model for partial differential equations, and it is only rather recently that it has come to be realized that the differences between the two theories are much more numerous and more profound than the
analogies.

*Dieudonné, Jean*, A panorama of pure mathematics (as seen by N. Bourbaki). Transl. from the French by I. G. Macdonald, Pure and Applied Mathematics, 97. New York etc.: Academic Press, a subsidiary of Harcourt Brace Jovanovich, Publishers. X, 289 p. (1982). ZBL0482.00003.

However, I'd like to mention at least a couple of very interesting surveys and a well-known graduate textbook:

*Brézis, Haïm; Browder, Felix*, **Partial differential equations in the 20th century**, Adv. Math. 135, No. 1, 76-144 (1998). ZBL0915.01011.
*Nirenberg, Louis*, Partial differential equations in the first half of the century, Pier, Jean-Paul (ed.), Development of mathematics 1900-1950. Based on a symposium organized by the Luxembourg Mathematical Society in June 1992, at Château Bourglinster, Luxembourg. Basel: Birkhäuser. 479-515 (1994). ZBL0807.01017.
*Evans, Lawrence C.*, Partial differential equations, Graduate Studies in Mathematics 19. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4974-3/hbk). xxi, 749 p. (2010). ZBL1194.35001.