This paper is devoted to studying the Cauchy problem for a family of dispersive–dissipative BO equations 𝑢𝑡+ℋ⁢𝑢xx−(𝐷𝛼𝑥−𝐷β𝑥)⁢𝑢+𝑢⁢𝑢𝑥=0. For a wide class of parameters β>1 and 0<𝛼<β, taking into account dispersive and dissipative effects, we establish sharp well-posedness results in Sobolev spaces 𝐻𝑠⁡(ℝ) and 𝐻𝑠⁡(𝕋) which yield new well-posedness conclusions for some physical relevant models.