I am learning differential geometry and have a few questions on curvature. -- Background:

1. Gauss invented "Gauss curvature" to measure how surface curves. 

2. Riemann gives an ingenious generalization of Gauss curvature from surface to higher 
   dimensional manifolds using the "Riemannian curvature tensor" (sectional curvature
   is exactly the Gauss curvature of the image of the "sectional" tangent 2-dimensional
   subspace under the exponential map).

3. In modern textbooks on differential geometry, people usually first define the notion
   of connection, and then the Riemannian curvature tensor is expressed in terms of
   connection.

My questions are:

1. Are there some other notions of "curvature", besides Gauss curvature and 
   Riemannian curvature tensor as its generalization, which people have invented to
   measure how space curves?

2. In history, who first introduced the notion of connection to describe
   the Riemannian curvature tensor, and why is this idea natural?

Thank you very much!