My mother-in-law passed away, not sure if there was a Will and there are no next of kin left, should I persue her estate?

First of all, if your mother-in-law left a Will its terms will control just how her estate will be distributed. If you are a beneficiary you will be notified. If you not, you can wait until it is filed in the probate court of the county in which she was domiciled as of her death. You can get a copy for a small fee since at that point it becomes a matter of public record.

If she had no Will then things are different. When someone dies without a Will they are said to have died "intestate". All states have enacted laws to deal with these type of situations and they provide for just how the deceased's estate will be divided; in other words, they determine who will inherit the property (heirs) and in what shares. Typically, an estate goes the spouse and children. If there are none, then to the next closest "kin" (i.e. blood relative).

If the deceased has no close living kin, there exist mechanisms to determine other blood relatives qualified to take the estate; no matter how far removed that they may be. The fact is that there is a strong preference to distribute the deceased’s property to blood relatives. However, in those rare cases where no living heir can be located, then the decedent’s estate will "escheat" to the state. This means that the state takes ownership of the decedent’s(your mother-in-laws) property (it will be the state of her domicile as of the date of her death).

Note: If you and your late husband had children, the they would be entitled to inherit their father's share of the estate (i.e. the share that he would have received if he were still alive).