Before revision:  Jumps, in particular exponential L\'evy and affine models, which exhibit small-maturity exploding smiles, have historically been proposed to remedy this (see Tankov for an overview) . A recent breakthrough was made by Gatheral, Jaisson and Rosenbaum, who proposed to replace the Brownian driver of the instantaneous volatility by a short-memory fractional Brownian motion, which is able to capture the short-maturity steepness while preserving path continuity .
After revision:  Jumps, in particular exponential L\'evy and affine models, which exhibit small-maturity exploding smiles, have historically been proposed to remedy this (see Tankov for an overview) , and more recently rough volatility models \mbox{%DIFAUXCMD AlosLeon, Fukasawa .