Underwater autonomous robotic operations require online localization and 3D mapping. Because of the absence of absolute positioning underwater, these tasks strongly rely on embedded sensors, including proprioceptive or navigation sensors - which can be fused for an odometry, - and exteroceptive sensors. One of the most popular exteroceptive sensors for underwater is the imaging sonar, which emits a large fan-shaped acoustic signal and estimates the position of the surrounding obstacles from a measure of the reflected signal. This paper addresses underwater online localization and 3D mapping using a forward looking, wide-aperture imaging sonar and vehicle's intrinsic navigation estimates. We introduce 3DSSDF (3D Sonar Reconstruction Using Signed Distance Functions), a new localization and 3D mapping algorithm based on signed distance functions, which is evaluated in simulation and on real data, in man-made and natural environments. Comparisons to reference trajectories and maps demonstrate that, in our tests, 3DSSDF efficiently corrects navigation drift and that trajectory and map accuracy is always below 1 m and below 1% of the distanced travelled, which can be sufficient for the safe inspection of natural or artificial underwater structures.