We present new infrared images and photometry with Spitzer/IRAC + MIPS of over 20 luminous compact blue galaxies (LCBGs), half at z 0 and half at z 0.75. LCBGs are extreme starburst galaxies with strong evolution, providing up to 40% of the observed increase in global star formation rate density between z=0 and z=1. They are important to study as they not only play a vital role in galaxy evolution, but they are also excellent local analogs for distant starburst galaxies, such as Lyman break galaxies at z>3. Almost all of our LCBGs are detected in IRAC, and most are also detected in MIPS 24um. We fit multi-component stellar population synthesis models to the full spectral energy distributions and set new constraints on both the underlying stellar populations in these low-mass, low-metallicity, high specific star formation rate-systems and on the presence of additional star formation hidden from the optical by obscuring dust. We combine the Spitzer data with high-resolution rest-UV and -optical imaging from HST to characterize the spatial distribution of star formation and to assess the incidence of major mergers driving the starbursts.