What does the path of an object look like when it is in uniform motion?
An object is having a uniform motion if it covers equal distance in equal interval of time (which implies speed is constant!). So the path can be straight or curved.
For eg. Consider a circular path. For understanding purposes, divide the circumference of the circle in six equal parts each subtending at the centre. The object covers each equal part in equal amount of time. Hence, by definition, this object is in uniform motion.