If your spouse had non-probate assets such as insurance or retirement benefits and you are the beneficiary, you typically get those assets paid out or transferred to you fairly quickly and simply. These typically do not need to go through probate if there is a surviving beneficiary designation.
If your spouse had “payable on death” or “right of survivor” designations on banking and other accounts, they will most likely pass to the surviving spouse.
If your spouse owned real property, it may depend on whether that real property was separate or community and whether your spouse had a valid Will or not.
If your spouse had a valid Will, they could convey their interest in any real property they own to the surviving spouse, whether separate or community. However, their interest, whether separate, or community, will go to the the beneficiary named in their Will. The Will should be probated to transfer title to the property to the beneficiary.
If your spouse did not have a valid Will, the person inheriting the real property will depend on whether your spouse had children with someone other than you and whether the real property is separate or community. If your spouse had no children, or no children with anybody else, you should inherit the real property. If your spouse had children with somebody else and owned separate property, the children will inherit the real property and you will inherit a life estate in one-third (1/3) of the real property (meaning that you can use it during your lifetime). If your spouse had children with somebody else and owned community property, the children will inherit your deceased spouse’s one-half (1/2) interest and you will retain ownership in your one-half (1/2) of the real property. If your spouse did not have a valid Will, an heirship determination and a probate administration should be completed in order to transfer the title of the property.
Here is a link to the Travis County Probate Court’s diagram of descent and distribution without a valid Will:
*The above is an overly simplified answer to the question. There are other scenarios where probate could pass outside of probate such as if the property is held in trust or if there is a transfer on death deed.