The castle is now the seat of the Court of Law. Earlier from 1821 until 1981 it was a prison. Today's building dates back to 1903-1905.
The first castle dates back to the 16th century and was ordered to be built by Mikolaj Spytek Ligeza. It had to be rebuilt and enlarged several times and in the 17th/18th century it was home to the Lubomirski family (Rzeszow line).
You may walk around the building on the ramparts. To the left of the main entrance there are some old cannons to be seen (picture 3).
I'm afraid I don't know what this building is -- maybe one of the locals can help? -- it's between the end of ul. 3 Maja and the Castle -- but it's a good example of the sort of architecture you can see when wandering the streets.
If you follow ul. 3 Maja away from the Rynek end, you'll eventually come to the Old Town Castle. It dates back to the end of the 16th century, but following recent renovation it looks brand new. In the past it's been used as a jail, but it's now the law courts.
The townhall was first erected in the 16th century. It was rebuilt many times since then. From 1897 till 1900 was the last big modification. The decorations are in pseudo-gothic style.
The main shopping street in Rzeszow is ul. 3 Maja.
In my photo, you can see the Piarist Church and former Monastery on the left -- this complex houses the Town Museum (closed Mondays).