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V.~Kac classified finite dimensional simple Lie superalgebras over the field of complex numbers in 1970s. They constitute several series and are divided into two types based on their even part: classical type and Cartan type. Lie superalgebras of Cartan type consists of four series $W(n)$, $S(n)$, $S^{~}(n)$, and $H(n)$. In this talk I will briefly introduce Cartan type Lie superalgebras and then I will show how explicitly one can compute cohomology and support varieties for $W(n)$ and $S(n)$.  I will also mention what we have for $H(n)$ and present a new theorem on realizability of support varieties that holds both for classical and Cartan type Lie superalgebras. If I have time I am planning to present some open problems at the end of my talk.  The results about $W(n)$ represents joint work with Jonathan Kujawa and Daniel Nakano.
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