Unbounded Geodesics and Nonpositive Curvature

Question

I have the following interesting(?) question:

Let $M$ be a complete Riemann manifold. Suppose that it is of non-negative curvature. I want to know if $M$ has unbounded geodesics.

As the question is stated right now, the answer is no - the flat torus is a good example. Therefore, we need to add some demands on $M$ for the answer to be yes.

1. Suppose we assume that the curvature on $M$ is not identically zero. Is there an unbounded geodesic on $M$?

2. Suppose that the curvature on $M$ is strictly negative. Are all geodesics on $M$ unbounded?

3. Suppose that $M$ is simply connected, and no additional assumption on the curvature of $M$ is made. Are some (all?) geodesics on $M$ unbounded?

4. Clear all the assumptions made in the previous subquestions, and assume that $M$ is non-compact. Are all geodesics on $M$ unbounded?

Edit: Just to be clear , an unbounded geodesic is a geodesic which cannot is not confined in any bounded set - e.g. straight lines and geodesics in the hyperbolic space. Thanks to user8268 for adding the completeness demand.