Spontaneous motion has attracted lots of attention in the last decades in fluid dynamics for its potential application to biological problems such as cell motility. Recently, several model experiments showing spontaneous motion driven by chemical reactions have been proposed and revealed the underlying mechanism of the motion. Accordingly, several simple theoretical models have been extensively studied such as active Brownian particles, squirmers, self-thermophoretic swimmers and so on. We theoretically derive a set of nonlinear equations showing a transition between stationary and motile states driven away from an equilibrium state due to chemical reactions. A particular focus is on how hydrodynamic flow destabilizes an isotropic distribution of a concentration field. It is of interest that due to self-propulsive motion and flow around the droplet, a spherical shape becomes unstable and it elongates perpendicular to the direction of motion. This implies that the self-propulsion driven by chemical reaction is characterized as a pusher in terms of a flow field.
Even for the simpler models of self-propelled particles and drops, our understandings of their interactions are still primitive. In particular, it is of interest how hydrodynamic interactions and the interaction mediated by a concentration field give rise to collective behaviors such as motility-induced phase separation, global polar state, and clustering. I shall argue these issues based on our recent results of theoretical calculations.