After briefly reviewing the role of the cosmological constant (Lambda) in our uni-verse and more generally in a multi-verse, the focus will be on the local effects of this second possible fundamental constant of gravity. Three different results will be touched upon. The first one discusses the motion of a test particle in Schwarzschild- de Sitter (Kottler) metric with the emphasis on how the scales of the Newtonian constant mix with Lambda to give new relevant quantities of astronomical order of magnitude. The second topic zeroes around the entrance of Lambda into the so-called Generalized Uncertainty Principle (GUP) and its role in Hawking evaporation of black holes. Here the emphasis will be on the emergence of maximum and minimum temperature in black hole radiation. Finally, the last effect of Lambda to be discussed is its impact on the propagation of gravitational waves. A critical distance due to Lambda is found beyond which the "wavy" character of the gravitational wave is lost.