We find a new class of (2,0)-supersymmeric two-dimensional sigma models with torsion and target spaces almost complex manifolds extending similar results for models with (2,2) supersymmetry. These models are invariant under a new symmetry which is generated by a Noether charge of Lorentz weight one and it is associated to the Nijenhuis tensor of the almost complex structure of the sigma model target manifold. We compute the Poisson bracket algebra of charges of the above (2,0) -and (2,2)-supersymmeric sigma models and show that it closes but it is not isomorphic to the standard (2,0) and (2,2) supersymmetry algebra, respectively. Examples of such (2,0)- and (2,2)-supersymmetric sigma models with target spaces group manifolds are also given. In addition, we study the quantisation of the (2,0)-supersymmetric sigma models, compute the anomalies of their classical symmetries and examine their cancellation. Furthermore, we examine the massive extension of (2,0)-supersymmetric sigma models with target spaces almost complex manifolds, and study the topological twist of the new supersymmetry algebras.