Poincar\'e inequalities are among the most studied inequalities in probability and functional analysis. In this talk we will discuss the $L_p$ Poincar\'e inequalities with constants $C\sqrt{p}$. They are weaker than the log-Sobolev inequality, but still imply subgaussian concentration and transportation cost inequalities. We use martingale methods and infinite dimensional Brownian motions to prove these inequalities in various contexts, and demonstrate that certain noncommutative techniques are helpful even in some commutative settings.