Explore the persistence and first-passage properties of stochastic processes in this comprehensive lecture by Satya N Majumdar at the International Centre for Theoretical Sciences. Delve into applications across diverse fields including chemical reactions, search processes, finance, and astrophysics. Examine key concepts such as first passage probability, Markov processes, and the run-and-tumble particle model. Investigate collective effects, Monte Carlo methods, and universal survival probability for multi-dimensional systems. Gain insights into nonequilibrium physics and its interdisciplinary relevance through this in-depth exploration of fluctuations in stochastic processes.