For any finite dimensional (complex) simple Lie superalgebra g, I present an explicit recipe to construct an algebraic supergroup G (in terms of its functor of points) whose tangent Lie superalgebra is the given g. This goes through a generalisation of the Chevalley's method.

Chevalley's original procedure constructs a (semi)simple algebraic group starting from any complex, f. d. (semi)simple Lie algebra g and a faithful f. d. g-module V. The key tools to make use of is given by suitable integral forms - of g, U(g) and V - defined out of the notion of "Chevalley basis" for g.
I shall show that this method can be succesfully adapted when one shifts from the classical setting to the "super" one, starting from a convenient notion of "Chevalley basis" for simple Lie superalgebras.
Besides this existence result, I shall prove uniqueness: every connected algebraic supergroup whose Lie superalgebra be (f.d.) simple is isomorphic to one of the supergroups constructed via the Chevalley procedure. This eventually yields a complete classification of such supergroups.

References:
R. Fioresi, F. Gavarini, "Chevalley Supergroups", Memoirs of the AMS 215 (2012), no. 1014, pp. 1-77
R. Fioresi, F. Gavarini, "On the construction of Chevalley Supergroups", in: S. Ferrara, R. Fioresi, V. S. Varadarajan (eds.), "Supersymmetry in Mathematics and Physics", UCLA Los Angeles, USA 2010, Lecture Notes in Mathematics, Vol. 2027, 2011, Springer & Verlag, Berlin-Heidelberg-New
York, pp. 101-123
R. Fioresi, F. Gavarini, "Algebraic supergroups with classical Lie superalgebras", Journal of Lie Group Theory 23 (2013) , no. 1, 143-158
F. Gavarini, "Chevalley Supergroups of type D(2,1;a)", Proceedings of the Edinburgh Mathematical Society (in press), 27 pages - see also arXiv:1006.0464 [math.RA], (2010)
F. Gavarini, "Algebraic supergroups of Cartan type", Forum Mathematicum (in press), 92 pages - see also arXiv:1109.0626 [math.RA] (2011)