Abstract: Monodromy of local systems is one of the most fascinating and mysterious objects in algebraic and arithmetic geometry. In these talks I will discuss monodromy in the world of p-adic geometry, both in the arithmetic and geometric contexts. In particular, I will survey some of the recent developments on the so-called p-adic monodromy conjecture for curves, which asserts that every de Rham p-adic local system on a smooth projective curve over a p-adic field becomes semistable after passing to a finite cover. This result is a relative version of the classical p-adic monodromy theorem, which asserts that every de Rham representation is potentially semistable. I will then discuss some applications of this result (and techniques used in the proof). Among which, I will address certain rigidity properties for crystalline local systems. If time permits, I will also discuss some aspects of geometric monodromy of p-adic local systems along normal crossing divisors and some applications. This talk is based on some joint works (including ongoing projects) with Hansheng Diao, as well as joint works with Hansheng Diao, Heng Du, Yong Suk Moon.