Let 1 < p 6= 2 < ∞ and let L p (M) be a noncommutative L p -space associated to some von Neumann algebra M. We say that a bounded c0-semigroup (Tt)t≥0 on L p (M) admits a group dilation if there is another noncommutative L p -space L p (N), a bounded c0-group (Ut)t on L p (N) and two bounded operators J : L p (M) → L p (N) and Q: L p (N) → L p (M) such that Tt = QUtJ for any t ≥ 0. One has a similar definition for a discrete semigroup (T n )n≥0 associated with a single operator T : L p (M) → L p (M). This talk will be devoted to the question of determining which semigroups (either continuous or discrete) admit a group dilation. We will review a few recent theoritical results, some important examples, and connections with H∞ functional calculus and square functions.