Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)The aim of this article is to explore the class of intermediate logics between the truth-preserving Lukasiewicz logic L and its degree-preserving companion L=. From a syntactical point of view, we introduce some families of inference rules (that generalize the explosion rule) that are admissible in L= and derivable in L and we characterize the corresponding intermediate logics. From a semantical point of view, we first consider the family of logics characterized by matrices defined by lattice filters in [0,1], but we show there are intermediate logics falling outside this family. Finally, we study the case of finite-valued Lukasiewicz logics where we axiomatize a large family of intermediate logics defined by families of matrices (A, F) such that A is a finite MV-algebra and F is a lattice filter.243288320FP7-PEOPLE-IRSES project MaToMUVI [PIRSES-GA-2009-247584]FAPESP [2010/51038-0]CNPq [PQ 308524/2014-4]MINECO [TIN2012-39348-C02-01]Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)