In this note, which is a continuation of a previous paper of the authors [Set-Valued Analysis 6 (1998) 302-312], we study two classes of maximal monotone operators on general Banach spaces which we call C₀ (resp. C₁)-regular. All maximal monotone operators on a reflexive Banach space, all subdifferential operators, and all maximal monotone operators with domain the whole space are C₁-regular and all linear maximal monotone operators are C₀-regular. We prove that the sum of a C₀ (or C₁)-regular maximal monotone operator with a maximal monotone operator which is locally inf bounded and whose domain is closed and convex is again maximal monotone provided that they satisfy a certain "dom--dom" condition. From this result one can obtain most of the known sum theorem type results in general Banach spaces. We also prove a local boundedness type result for pairs of monotone operators.